準周期構造の理論的基礎 Theoretical introduction to...

準周期構造の理論的基礎Theoretical introduction to

quasiperiodic structures

Nobuhisa Fujita

IMRAM Tohoku University

Sendai 980-8577 Japan

第一回新学術領域ハイパーマテリアル若手研究会Web開催2020５28

1

Outline

1 Introduction pp3-8

2 Z-modules for quasicrystals pp9-23

3 Quasiperiodic tilings (QPTs) pp24-31

4 Methods for generating QPTs pp32-61

5 Approximants pp62-78


― 準周期構造の理論と基礎 ―

2

1 Introduction

ED pattern from a rapidly quenched Al-Mn alloy

D Shechtman et al Phys Rev Lett 53 (1984) 1951mdash1953



3


5-fold rotational symmetry is not compatible with periodicity still the diffraction peaks are very sharp


― 準周期構造の理論と基礎 ―1 Introduction

4

Quasicrystals as a new class of ordered solids

D Levine and PJ Steinhardt Phys Rev Lett 53 (1984) 2477mdash2480

1 Long-range quasiperiodic translational order

A QC exhibits a self-similar arrangement of Bragg peaks (120633functions) whose indexing needs 119896 (gt 119889) independent basis vectors for indexing (119889 the number of space dimensions)

2 Non-crystallographic point group symmetry

A QC exhibits a point group symmetry forbidden in periodic crystals (eg 119899-fold rotational axes with 119899 being a natural number excluding 1 2 3 4 and 6)



5

Paradigm shift in crystallography the New definition of crystals (1992)

International Union of Crystallography Report of the Executive Committee for 1991 Acta Cryst A48 (1992) 922mdash946

The definition proposed by the IUCr Commission on Aperiodic Structures (IUCr 1991)

by crystal we mean any solid having an essentially discrete diffraction diagram and

aperiodic crystal we mean any crystal in which three-dimensional lattice periodicity

can be considered to be absent



6

Aperiodic order Three known categoriesQuasiperiodic structure (QCs incommensurate modulation composite crystals)

A structure that show Bragg peaks (120633 functions) in diffraction patterns which require 119896 (gt 119889) independent reciprocal basis vectors for indexing where 119889 is the of space dimensions and 119896 lt infin

Limit-periodic structureA structure with recursive (or hierarchical) superlattice structures superposed on a basic periodic latticeK Niizeki and N Fujita Philos Mag 87 (2007) 3073mdash3078 and references cited therein

Limit-quasiperiodic structureA structure that can be obtained as an incommensurate section of a limit-periodic structure in higher dimensionsK Niizeki and N Fujita J Phys A Math Gen 38 (2005) L199mdashL204 and references cited therein



7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim

Metallic alloysSoft matter Metallic

alloys

Non-crystallographic point groups for quasicrystals



8

2 Z-Modules for quasicrystals

Reciprocal (= Fourier) module 120028lowast

≃ A dense (ne discrete) point set in the 119941-dim wave-number (or reciprocal) space generated as the integer linear combinations of 119948 (gt 119941) reciprocal basis vectors used to index the Bragg peaks

Direct (= Bravais) module 120028

≃ A dense (ne discrete) point set in the 119941-dim real (or direct) space generated as the integer linear combinations of 119948 (gt 119941) basis vectors used to index the quasi-lattice points or tiling vertices

Additive Abelian group over an integer ring (Z) Alternatives to lattices for periodic crystals



9

(Z-加群)

( 119948 rank 119941 space dimensions )

Ex) Fourier module for decagonal QC


― 準周期構造の理論と基礎 ―2 Z-modules for quasicrystals

10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm

Minimal basis set for indexing the Bragg peaks



11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups

octakaidecagonaldodecagonal

decagonaloctagonalpentagonalicosahedral

5m 8mm 10mm 12mm 18mm ത5ത32119898

For a planar 119899-gonal quasicrystal the minimal number of basis vectors is 119896 = 120601 119899 where 120601 119899(Eulerrsquos 120601-function) is the number of positive integers up to 119899 that are co-prime with 119899

120601 119899 = 119899ς119895 1 minus1

119901119895(119901119895 isin all prime factors of 119899)

σ119895120601 119889119895 = 119899 (119889119895 isin all divisors of 119899)

1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899

Check There are 119899 minus 120601 119899 independent constraints for the 119899 unit vectors



12

Lifting up the of space dimensions

119889-dim Fourier module ℳlowast of rank 119896can be lift up to 119896-dim reciprocal hyper-lattice ℒlowast

Hyper-space119896-dim

ℳlowastReciprocal space

(perpendicular dims)(119896 minus 119889)-dim Reciprocal space

(physical dims)119889-dim

Observed Bragg peaks

119896 basis vectors



13





14

ℒlowast

The amplitude of each Bragg peak in the 119889-dimensional Fourier space is assigned to the corresponding reciprocal hyper-lattice point in 119896 dimensions

Reciprocal space(perpendicular dims)

(119896 minus 119889)-dim Reciprocal space(physical dims)

119889-dim

Complementary components 119944119895perp

119866ℳlowast point group of the Fourier module ℳlowast of the QC

119866ℒlowast point group of the 119896-dim reciprocal hyper-lattice ℒlowast

119866ℒlowast should have a subgroup 119867 which is isomorphic to 119866ℳlowast (ie 119866ℒlowast sup 119867 cong 119866ℳlowast) and which does not mix the physical and orthogonal components ie

ഥ119863 =119863 00 119863perp where ഥ119863 isin 119867 and 119863 isin 119866ℳlowast



15

Lifting up the of space dimensions[119896-dim basis vectors ഥ119944119947 = (119944119947 119944119947

perp) of ℒlowast]



16

[1] K Niizeki J Phys Math Gen 22 (1989) 193mdash204[2] A Yamamoto Acta Cryst A52 (1996) 509mdash560

+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components

Perpendicular components

5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955

Fourier tr(119948-dim)


projection section

(Direct space)(Reciprocal space)



Physical dims(d-dim)

Hyper-space(k-dim)

17

lt

ℒ

The real charge density in the physical space is a 119889-dim section of a periodic array of atomic surfaces in 119896-dim hyper-space obtained as the Fourier transform of the 119896-dim structure factors

Charge density (periodic in hyper-space)

Direct space(perpendicular dims)

(119896 minus 119889)-dim Direct space(physical dims)

119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895

119896-dim bases ഥ119938119895 of the direct hyper-lattice ℒ= Fourier tr of the reciprocal hyper-lattice ℒlowast



19

119896-dimensional bases ഥ119938119895 = (119938119895 119938119895perp) of ℒ



20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845

hyper-lattices for quasicrystals

bull2-dim QC A unique hyper-lattice exists for 119899-gonal case

(119899 = 5 8 10 12 18 hellip)

bull3-dim QC Three hyper-lattices exist for icosahedral case P-type (SI) F-type (FCI) I-type (BCI)

ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21

Direct (= Bravais) module ℳ(projection of ℒ onto the physical sub-space)

Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)

Perpendicular dims ((k-d)-dim)

Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22

Self-similarity of ℳ (and ℳlowast)Z-modules for quasicrystals are scale invariant There exists an irrational number 120649 called the Pisot

unit such that 120591ℳ=ℳ and 120591 gt 1

Pisot unit for important classes of QCs

ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787

[1] DS Rokhsar et al Phys Rev B 35 (1987)5487mdash5495[2] LS Levitov and J Rhyner J Phys France 49 (1988) 1835mdash1849[3] K Niizeki J Phys Math Gen 22 (1989) 193mdash204

[12] [12][3] [3] [3] [3]



23

3 Quasiperiodic tilings (QPTs)

P1 tiling

R Penrose Math Intel 2 (1979) 32ndash37 B Gruenbaum GC Shephard Tilings and Patterns (Freeman New York 1987) Chap 10



24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling


― 準周期構造の理論と基礎 ―3 Quasiperiodic tilings (QPTs)


72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus

Ammann-Beenker tiling (8mm)



27

F P M Beenker Eindhoven University of Technology Report No 82-WSK-04 (Eindhoven 1982)

Square+

Equilateral triangle

Stampfli tiling(12mm)



28

P Stampfli Helv Phys Acta 59 (1986) 1260ndash1263J Hermisson C Richard M Baake J Phys I France 7 (1997) 1003ndash1018

Fibonacci chain (or tiling) (1-dim)

The 119909 coordinate of every lattice point is represented as 119909 = 119894 +119895

120591

which belongs to the 1-dim Z-module ℳ = 119898+ 119899120591 | 119898 119899 isin ℤ

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)

L L L L L L L LS S S S S

Lrsquo Srsquo Lrsquo Lrsquo Srsquo Lrsquo Srsquo Lrsquo

LSLLSLSLLSLLShellip cong LrsquoSrsquoLrsquoLrsquoSrsquoLrsquoSrsquoLrsquohellip (Lrsquo=LS Srsquo=L) Cf 120591ℳ=ℳ



29


Perpendiculardims 119909perp

Physical dims 119909∥

ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30

Cut-and-project methodS

pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31

4 Methods for generating QPTs

Cut-and-projection

Section method

Substitution method(inflation-deflation)

Dual-grid method

(NG de Bruijn 1981)

Methods for constructing QPTs



32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1


― 準周期構造の理論と基礎 ―4 Methods for generating QPTs

33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34

Cut-and-project method≃ Section method

Sp

an o

f w

ind

ow

119909perp

119909∥

Check An atomic surface intersect with the physical space if and only if the corresponding lattice point is within the strip

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35

Dual-grid method≃ Cut-and-project method ≃ Section method



36

Check All the squares which intersect with the horizontal axis are colored in red The left base points of the red square are inside the strip

119909∥

119909perp




37

119909∥

119909perp

The base points of all the red squares are projected down to the physical space to form the vertices of the aperiodic tiling ndash the dual-grid method

L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1

Self-similarity ndash an geometrical basis

If we remove the vertex between L and S in every LS pair we get a new segment Lrsquo This is done simply by narrowing the window

[Cut-and-project method]

119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo

Here the window is reduced by a factor of 1120591 The new Lrsquo tile corresponds to the projection of a diagonal of the square unit cell



39

Self-similarity ndash an geometrical basis[Cut-and-project method]

119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591

If the unit cell is chosen as a parallelogram as shown the new tiles Lrsquo and Srsquo correspond to the projection of the edges of the parallelogram



40


119909prime∥

119909primeperp

Lrsquo Srsquo

The Affine transformation ҧ119860 which rescales the 119909∥ and 119909perp axes by 1120591 and minus 120591 respectively will recover the original setting (cut-and-projection)

ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41

Dual-grid method ndash practice in 1 dim

Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)

Grid lines(1st series)

Grid lines(2nd series)

One-to-one correspondence

Intersecting cells(red squares)



42

Two-dimensional grid lines (in hyper-space)

1198991+ 1199041 ഥ1199381 + 120579ഥ1199382 (minusinfin lt 120579 lt infin) hellip 1st series

120579ഥ1199381 + 1198992+ 1199042 ഥ1199382 (minusinfin lt 120579 lt infin) hellip 2nd series

( 1199041 1199042 hellip phason shift parameters )

One-dimensional grid points (in physical sub-space)

(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)

rarr 119909 = (2 minus 1120591)(1198991+ 1199041) hellip 1st series

120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)

rarr 119909 = (1 + 2120591)(1198992+ 1199042) hellip 2nd series

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43

Base points of the two unit cells that contact each other at a grid point (in the physical space)

(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)

n2 = 120579 minus Frac[120579] = Floor[120579]

(n1 n2) (n1-1 n2) hellip indices for the squares

(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)


(n1 n2) (n1 n2-1) hellip indices for the squares

The vertices of the tiling are the projections of these base pointsbase point (n1 n2) harr vertex coordinate x = n1 + n2 120591




44

Dual-grid method ndash practice in 2 dims

Embedding dimension = 5

(119909 119910 119911 119906 119907)physical perpendicular

space space

1199381∥

edge vectors inthe physical space

1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345

Check ഥ119938119946 sdot ഥ119938119947 = 120633119946119947 hellip the Kronecker delta

physical space perpendicular space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46

Dual-grid method ndash practice in 2 dims5-dim hyper-cubic lattice ℒ = 1198991ഥ1199381+ 1198992ഥ1199382+ 1198993ഥ1199383+ 1198994ഥ1199384+ 1198995ഥ1199385

Hyper-grids in 5-dim hyper-space1st grids (1198991+ 1199041 )ഥ1199381 + 1205792ഥ1199382+ 1205793ഥ1199383+ 1205794ഥ1199384+ 1205795ഥ11993852nd grids 1198992+ 1199042 ഥ1199382 + 1205791ഥ1199381+ 1205793ഥ1199383+ 1205794ഥ1199384+ 1205795ഥ1199385

3rd grids (1198993+ 1199043 )ഥ1199383 + 1205791ഥ1199381+ 1205792ഥ1199382+ 1205794ഥ1199384+ 1205795ഥ11993854th grids (1198994+ 1199044 )ഥ1199384 + 1205791ഥ1199381+ 1205792ഥ1199382+ 1205793ഥ1199383+ 1205795ഥ11993855th grids (1198995+ 1199045 )ഥ1199385 + 1205791ഥ1199381+ 1205792ഥ1199382+ 1205793ഥ1199383+ 1205794ഥ1199384

2-dim grid lines (the physical-space sections of the hyper-grids)

1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )



(120579119895 free parameter)


47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2

an intersection of grid lines




49

Dual-grid method ndash practice in 2 dims(119909 119910 0 0 0) = (1198991+ 1199041)ഥ1199381+ (1198992+ 1199042)ഥ1199382+ 1205793ഥ1199383+ 1205794ഥ1199384+ 1205795ഥ1199385

1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)

1205794 = ഥ1199444 119909 119910 0 0 0 = 1198994 + 1199044+ 120575rsquo 1198994 = 1205794 minus 1199044 120575rsquo = 119865119903119886119888(1205794minus 1199044)

1205795 = ഥ1199445 119909 119910 0 0 0 = 1198995 + 1199045+ 120575rsquorsquo 1198995 = 1205795 minus 1199045 120575rsquorsquo = 119865119903119886119888(1205795minus 1199045)

where ഥ119944119895 = ഥ119938119895 ( 119895 = 1 2 3 4 5) hellip 5-dim reciprocal basis vecs

These equations determine five integers 1198991 1198992 1198993 1198994 1198995 which provide the indices of the base points of the four 5-dim unit cells that contact each other at the above intersection point (xy) of the 1st and 2nd grid lines rarr

① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)

③ 1198991 1198992minus 1 1198993 1198994 1198995④ (1198991minus 1 1198992minus 1 1198993 1198994 1198995)

(similar formulas can be obtained foran intersection of ith and jth grid lines)



50


(119909 119910 0 0 0)

①②

③④

1198991

1198992

Each intersection of two grid lines gives one

rhombus in the tiling



51




Outer boundary (rhombic icosahedron)Projection of a 5-dim hyper-cubic unit cell into the 3-dim perpendicular space

Rhombic Penrose tiling (in 2-dim physical space) Mapping the vertices of the tiling intothe orthogonal complement (3-dim)

52

P3 tiling


(119909 119910 119911 119906)


Dual-grid method ndash practice in 2 dimsAmmann-Beenker tiling


45˚

1199384∥ 1199383

∥

1199382∥

1199381∥

physical perpendicular

space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54

4-dim hyper-cubic lattice ℒ = 1198991ഥ1199381+ 1198992ഥ1199382+ 1198993ഥ1199383+ 1198994ഥ1199384

Hyper-grids in 4-dim hyper-space1st grids (1198991+ 1199041 )ഥ1199381 + 1205792ഥ1199382+ 1205793ഥ1199383+ 1205794ഥ11993842nd grids 1198992 + 1199042 ഥ1199382 + 1205791ഥ1199381+ 1205793ഥ1199383+ 1205794ഥ1199384

3rd grids (1198993+ 1199043 )ഥ1199383 + 1205791ഥ1199381+ 1205792ഥ1199382+ 1205794ഥ11993844th grids (1198994+ 1199044 )ഥ1199384 + 1205791ഥ1199381+ 1205792ഥ1199382+ 1205793ഥ1199383


1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)

1205794 = ഥ1199444 119909 119910 0 0 = 1198994 + 1199044+ 120575rsquo 1198994 = 1205794 minus 1199044 120575rsquo = 119865119903119886119888(1205794minus 1199044)

where ഥ119944119895 = ഥ119938119895 ( 119895 = 1 2 3 4) hellip 4-dim reciprocal basis vecs




These equations determine four integers 1198991 1198992 1198993 1198994 which provide the indices of the base points of the four 4-dim unit cells that contact each other at the intersection (xy) of the 1st and 2nd grid lines rarr


① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)

③ 1198991 1198992minus 1 1198993 1198994④ (1198991minus 1 1198992minus 1 1198993 1198994 )

58

(x y 0 0)

②

③④

1198991

1198992

①




Ammann-Beenkertiling (8mm)

Each intersection of two grid lines gives one rhombus or

square in the tiling

59




Ammann-Beenker tiling (in 2-dim physical space) Mapping the vertices of the tiling intothe orthogonal complement (2-dim)

Outer boundary (regular octagon) Projection of the 4-dim hyper-cubic unit cell into the 2-dim perpendicular space

60

Perl scripts for constructing the rhombic Penrose tiling (P3) and the Ammann-Beenker tiling

Tilings encyclopedia

Book

httpwwwtagentohokuacjplabotsainobuhisaAmmannBeenker2povpl

httpwwwtagentohokuacjplabotsainobuhisaPenrose2povpl1199041 + 1199042+ 1199043 + 1199044+ 1199045 = integer rarr rhombic Penrose tiling (P3)1199041 + 1199042+ 1199043 + 1199044+ 1199045 = half integer rarr anti-Penrose tiling

Some resources on aperiodic tilings

httptilingsmathuni-bielefeldde

B Gruenbaum GC Shephard Tilings and Patterns W H Freeman and Company New York 1987 (Chapter 10)



61

5 Approximants[Cut-and-project method]

119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w

Fibonacci chain (without linear phason strain)



62

L L L L L L L LS S S S S L S L L S L L S


119909∥

119909perp

32 approximant to Fibonacci chain

Phason strain119878 = Δℓ

L S L L S L L LL S L S S L S L L S L SWin

do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1

L S L L S L LL S L S L S L L S L LL S



64

ℓ

Δ

Rational approximations of 120591

119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+

Continued fraction expansion of τ =1+ 5

2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655

Rational approximation of τ τ cong119865119899+1

119865119899

Fibonacci numbers


― 準周期構造の理論と基礎 ―5 Approximants

65

1199385||minus 120591(1199381

||+ 1199384

||) = 120782 xminusdirection

1199381||minus 1199384

||minus 120591(1199382

||minus 1199383

||) = 120782 yminusdir

physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384

perp = 120782 zminusdir

1199381perp minus 1199384

perp +1

120591(1199382

perp minus 1199383perp) = 120782 uminusdir

perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp

5-dim basis vectors of the P3 tiling(with no linear phason strain)



66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation



5-dim basis vectors of an approximant(modified with a linear phason strain)

67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912

3 minus 1205912119901 minus 119902minus119903

1 2

1199383 =

minus1205912

minus 3 minus 1205912119901 minus 119902119903

1 2

1199384 =

1(2120591)

minus 4120591 + 3(2120591)minus119901minus119904

1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68

Periodicity of 119901

119902119903

119904approximant

A general 5-dim lattice vector with indices ℎ1 ℎ2 ℎ3 ℎ4 ℎ5

rarr Perpendicular space components

ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874

perp + ℎ51198875perp

=

2119902ℎ5 minus 119901ℎ1 + (119901 minus 119902)ℎ2 + (119901 minus 119902)ℎ3 minus 119901ℎ4119904ℎ1 minus 119903ℎ2 + 119903ℎ3 minus 119904ℎ4

(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2

Note that the 2-dim lattice basis vectors 1199291 and 1199292 of 119901

119902119903

119904

approximant along the 119909 and 119910 directions should in general be indexed as 119895 119896 119896 119895 119894 and 119897 119898minus119898minus119897 0 respectively



69


119902119903

119904approximant

Their modified perpendicular space components would then be

1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =

2119901 119896 minus 119895 minus 119902 119896 minus 119894 0

(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873

perp minus 1198971198874perp =

02(119904119897 minus 119903119898)

0

These perpendicular space components would vanish if 1199291 and 1199292 are the lattice bases of the approximant so that

119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant

There exists non-zero integer 119899 such that

119894 minus 119896 = 119901119899 119895 minus 119896 = 119902119899 and5119896 = minus 119901 + 2119902 119899

Thus the values of 119894 119895 and 119896 are determined uniquely (up to a sign) according to the values of 119901 and 119902 and so are the values 119897 and 119898according to the values of 119903 and 119904 through

119897 = plusmn119903 119898 = plusmn119904

p+2qと5の最小公倍数を与えるようにnを決める



71


119902119903

119904approximant

Lattice basis vectors in the physical space

1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||

The corresponding increments in the perpendicular space

1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383

perp minus 1199031199384perp

The linear phason strain tensor 119878 (Definition 119961perp~119878119961∥)

1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292

perp 1199291 1199292minus1 NB) 119930 is a 3x2 matrix in the present case



72

How to generate an approximant with dual-grids



73

Replace the hyper-cubic lattice basis vectors (ഥ119938119895 and ഥ119944119895)

into modified basis vectors (119938119895 and 119944119895) while performing

the dual-grid method (see p50)

ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp

ഥ119944119895 (def ഥ119938119895 sdot ഥ119944119895 = 120633119894119895) rarr 119944119895 (def 119938119895 sdot 119944119895= 120633119894119895)

Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74

Examples Approximant No2

1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises

a Compare approximant tilings generated with different values of phason shifts

b Apply the dual-grid method for generating Ammann-Beenker tiling (octagonal or 8-gonal QC)

c Generate a few simplest approximants to the Ammann-Beenker tiling



79

Outline

1 Introduction pp3-8

2 Z-modules for quasicrystals pp9-23

3 Quasiperiodic tilings (QPTs) pp24-31

4 Methods for generating QPTs pp32-61

5 Approximants pp62-78



2

1 Introduction





3





4









5









6







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

1 Introduction





3





4









5









6







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79





4









5









6







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79









5









6







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79









6







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79







7

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

5m 8mm 10mm 12mm 18mm ത5ത32119898

2-dim 3-dim


alloys




8









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79









9

(Z-加群)





10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




10

ℳ10lowast = σ119895=1

4 119899119895119944119895 119899119895 isin ℤ

120591 =1 + 5

2= 2cos(

120587

5)

(golden mean)

Point group

1

t

1199441

1199442

1199443

1199444 (119944120787)

10mm




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




11

119889 = 2119896 = 4

119889 = 2119896 = 6

119889 = 3119896 = 6

g 6

g 5

g 4

g 2

g 3

g 1

Point groups



5m 8mm 10mm 12mm 18mm ത5ത32119898


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


120601 119899 = 119899ς119895 1 minus1



1199441

1199442

1199443

1199444

119944119899

119944120584 + 119944119889119895+120584 + 1199442119889119895+120584 +⋯+ 119944119899minus119889119895+120584 = 0 | 1 le 120584 le 120601(119889119895)1le119889119895lt119899




12











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79











13





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79





14

ℒlowast




119889-dim








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79








15


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


perp) of ℒlowast]



16


+ + ++

g 6

g 5

g 4

g 2

g 3

g 1

ḡ 5

ḡ 4

ḡ 2

ḡ 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Structure factors

119878(119918)| 119918 isin ℳlowast

ҧ119878(ഥ119918)| ഥ119918 isin ℒlowast

120588119889 119955 =

119918isinℳlowast

119878(119918)119890119894119918sdot119955

120588119896 ത119955 = ഥ119918isinℒlowast

ҧ119878(ഥ119918)119890119894ഥ119918sdotത119955



projection section





Hyper-space(k-dim)

17

lt

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

ℒ





119889-dim



18

120588119889 119955 = 120588119896 ത119955 |ത119955=(119955120782)

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

119861 =

ഥ1199441ഥ1199442

⋮ഥ119944119899

119860 =

ഥ1199381ഥ1199382⋮ഥ119938119899

= (119861minus1)119879

ഥ119938119894 ∙ ഥ119944119895 = 120633119894119895




19




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




20

+ + +

+

a 6

a 5

a 4

a 2

a 3

a 1

ā 5

ā 4

ā 2

ā 1

++

Physical components


5m 8mm 10mm 12mm 18mm ത5ത32119898

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

119811119842119851119838119836119853119825119838119836119842119849119851119848119836119834119845



(119899 = 5 8 10 12 18 hellip)


ℒ119875 ≔ 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866

ℒ119865 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 σ119895=16 119899119895 = 0 mod 2

ℒ119868 = 1198991 ത1198861 + 1198992 ത1198862 + 1198993 ത1198863 + 1198994 ത1198864 + 1198995 ത1198865 + 1198996 ത1198866 119899119894 = 119899119895 mod 2



21


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


Df) ℳ = σ119895=1119896 119899119895119938119895 119899119895 isin ℤ


Hyper-space(k-dim)


Direct space

Reciprocal space

ℳ

ℳlowast ℒlowast

ℒ

ℳperplowast

ℳperp



22




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




ℳ5

ℳ10

120783 + 120787

120784

ℳ8

120783 + 120784

ℳ12

120784 + 120785

ℳ18

120649120783 = 120783 + 120647 + 120647minus120783

120649120784 = 120649120783 120649120783 minus 120783

(120647 = 119942120645119946120791 )

ℳ119868hF

ℳ119868hI

120783 + 120787

120784

ℳ119868hP

120784 + 120787


[12] [12][3] [3] [3] [3]



23


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


P1 tiling




24

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

72deg

72deg

72deg

144deg

72deg

36deg

36deg

216deg

kite

dart

P2 tiling




72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

72degRhombus

+

36degRhombus

P3 tiling




Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Square+

45degRhombus




27


Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Square+





28




120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79



120591


1 1 1 1 1 1 1 11t 1t 1t 1t 1t

119909

120591 =1+ 5

2(golden mean)






29




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381 ഥ1199381ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382 ഥ1199382

Zigzag chain

1 1 1 1 1 1 1 11t 1t 1t 1t 1t

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



30


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


pa

n o

f w

ind

ow

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1

119909perp

119909∥



31


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


Cut-and-projection

Section method


Dual-grid method

(NG de Bruijn 1981)




32


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


pa

n o

f w

ind

ow

119909perp

119909∥

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



33

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Section methodS

pa

n o

f w

ind

ow

119909perp

119909∥

Atomic surface

ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



34


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


Sp

an o

f w

ind

ow

119909perp

119909∥


ഥ1199381 = 1minus1

120591 ഥ1199382 =

1

120591 1



35




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




36


119909∥

119909perp




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




37

119909∥

119909perp


L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

L S

ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




119909∥

119909perp



38


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


L S


119909∥

119909perp

ഥ1199381 + ഥ1199382 = 1205911

1205912 ഥ1199381 = 1 minus

1

120591

Lrsquo Srsquo




39


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119909∥

119909perp

Lrsquo Srsquo

ഥ1199391 = 1205911

1205912 ഥ1199392 = 1 minus

1

120591




40


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119909prime∥

119909primeperp

Lrsquo Srsquo


ҧ119860 ∙ ഥ1199391 = 1 minus1

120591 ҧ119860 ∙ ഥ1199392 =

1

120591 1 ҧ119860 =

1120591 00 minus120591



41


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


Grid points

1st series

2nd series

n1= -1 0 1 2 3 4 5 6

n2= 0 1 2 3

4

5

6

7

(22)

(12)

(11)

(23)(33)

(43)

(44)

(54)

(64)

(65)







42






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79






(1198991+ 1199041)ഥ1199381 + 120579ഥ1199382 = (119909 0) rarr 120579 = 1120591(1198991+ 1199041)


120579ഥ1199381 + (1198992+ 1199042)ഥ1199382 = (119909 0) rarr 120579 = 120591 (1198992+ 1199042)


ഥ1199381 = 1 minus1

120591 ഥ1199382 =

1

120591 1




43


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


(1st series) x = ( 2 - 1120591 ) (n1+s1) 120579 = 1120591(n1+s1)



(2nd series) x = ( 1 + 2120591) (n2+s2) 120579 = 120591(n2+s2)







44




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




space space

1199381∥


1199382∥

1199383∥

1199384∥

1199385∥

P3 tiling



45


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


ഥ119938119895 = 25 cos 120593119895 sin 120593119895 cos 2120593119895 sin 2120593119895 12

where 120593119895 =2120587119895

5 119895 = 12345



1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp



46





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79





1st grids 52(1198991 + 1199041 )1199381∥ + 120579(1199382

∥ minus 1199385∥

2nd grids 52(1198992 + 1199042 )1199382∥ + 120579(1199383

∥ minus 1199381∥)

3rd grids 52(1198993 + 1199043 )1199383∥ + 120579(1199384

∥ minus 1199382∥ )

4th grids 52(1198994 + 1199044 )1199384∥ + 120579(1199385

∥ minus 1199383∥ )

5th grids 52(1198995 + 1199045 )1199385∥ + 120579(1199381

∥ minus 1199384∥ )





47

Phason shifts

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2




48

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

1198991 1198991+ 11198991 minus 1

1198992+ 1

1198992

1198992minus 1

(119909 119910 0 0 0)

1st grids

2nd grids

1198992+ 2

1198991+ 21198991 minus 2





49


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


1205793 = ഥ1199443 119909 119910 0 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)





① (1198991 1198992 1198993 1198994 1198995 )

② (1198991minus 1 1198992 1198993 1198994 1198995)





50


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


(119909 119910 0 0 0)

①②

③④

1198991

1198992





51






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79






52

P3 tiling


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


(119909 119910 119911 119906)




45˚

1199384∥ 1199383

∥

1199382∥

1199381∥


space space


53



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79



ഥ119938119895 = 1 2 cos(120593119895) sin(120593119895) cos(3120593119895) sin(3120593119895)

where 120593119895 =120587(119895minus1)

4 119895 = 1234


45˚ 1199381perp

perpendicular space

119911

119906

1199382perp

1199383perp

1199384perp

45˚ 1199381∥

physical space

119909

119910

1199382∥

1199383∥

1199384∥


54





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79





1st grids 2(1198991 + 1199041 )1199381∥ + 1205791199383

∥

2nd grids 2(1198992 + 1199042 )1199382∥ + 1205791199384

∥

3rd grids 2(1198993 + 1199043 )1199383∥ + 1205791199381

∥

4th grids 2(1198994 + 1199044 )1199384∥ + 1205791199382

∥






55

Phason shifts



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79



1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2


56



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79



(x y 0 0)

1st grids

2nd grids

1198992 + 1

11989921198992minus 1

1198992+ 2

1198991 1198991+ 11198991 minus 11198991 minus 2



57

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

(119909 119910 0 0 0) = (1198991+ 1199041)ത1198861+ (1198992+ 1199042)ത1198862 + 1205793ത1198863+ 1205794ത1198864

1205793 = ഥ1199443 119909 119910 0 0 = 1198993+ 1199043 + 120575 1198993 = 1205793minus 1199043 120575 = 119865119903119886119888(1205793minus 1199043)








① (1198991 1198992 1198993 1198994 )

② (1198991minus 1 1198992 1198993 1198994)


58

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

(x y 0 0)

②

③④

1198991

1198992

①







59






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79






60



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79



Book








61


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119909∥

119909perp

ഥ1199381 = 1 minus120783

120649 ഥ1199382 =

1

120591 1

Win

do

w




62



119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119909∥

119909perp




do

w

ഥ1199381 = 1 minus120784

120785 ഥ1199382 =

1

120591 1



63

ℓ

Δ


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119909∥

119909perp



Win

do

w

ഥ1199381 = 1 minus120785

120787 ഥ1199382 =

1

120591 1




64

ℓ

Δ


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119865119899+1 = 119865119899 + 119865119899minus1 (1198650 = 0 1198651 = 1 1198652 = 1 1198653 = 2 1198654 = 3 1198655 = 5 1198656 = 8hellip )

120591 =1 + 5

2= 1 +

1

120591= 1 +

1

1 +1

1 +1

1 +1

1+


2(golden mean)

120591 cong 1 +1

1 +1

1 +1

1 + 1

= 1 +1

1 +1

1 +11986521198653

= 1 +1

1 +11986531198654

= 1 +11986541198655

=11986561198655


119865119899

Fibonacci numbers



65

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199385perp +

1

1205911199381perp + 1199384


1199381perp minus 1199384

perp +1

120591(1199382


perpendicular space

119911

119907 119906

1199381perp

1199382perp

1199383perp

1199384perp

1199385perp




66

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

1199385||minus 120591(1199381

||+ 1199384


1199381||minus 1199384

||minus 120591(1199382

||minus 1199383


physical space

1199381∥

1199382∥

1199383∥

1199384∥

1199385∥

119909

119910

1199395perp +

119902

1199011199391perp + 1199394


1199391perp minus 1199394

perp +119904

119903(1199392


perpendicular space

119911

119907 119906

1199391perp

1199392perp

1199393perp

1199394perp

1199395perp

119901

119902119903

119904approximation




67


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


1199381 =

1(2120591)

4120591 + 3(2120591)minus119901119904

1 2

1199382 =

minus1205912


1 2

1199383 =

minus1205912


1 2

1199384 =

1(2120591)


1 2

1199385 =

1021199020

1 2

119938119895 =

119938119895||

119939119895perp

1 2

NB z and u axes are

arbitrarily scaled

119901

119902119903

119904approximation



68


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119902119903

119904approximant



ℎ11198871perp + ℎ21198872

perp + ℎ31198873perp + ℎ41198874


=


(ℎ1 + ℎ2 + ℎ3 + ℎ4 + ℎ5) 2


119902119903

119904




69


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119902119903

119904approximant


1199291prime = 1198951198871

perp + 1198961198872perp + 1198961198873

perp + 1198951198874perp + 1198941198875

perp =


(119894 + 2119895 + 2119896) 2

1199292prime = 1198971198871

perp +1198981198872perp minus1198981198873


02(119904119897 minus 119903119898)

0


119901

119902=119896 minus 119894

119896 minus 119895

119903

119904=

119897

119898 119894 + 2119895 + 2119896 = 0

(119894 minus 119896) + 2(119895 minus 119896) + 5119896 = 0



70


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119902119903

119904approximant




119897 = plusmn119903 119898 = plusmn119904




71


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


119902119903

119904approximant


1199291 = 1198951199381||+ 1198961199382

||+ 1198961199383

||+ 1198951199384

||+ 1198941199385

||= 1199021198991199381

||+ 1199021198991199384

||+ 1199011198991199385

||

1199292 = 1199031199381||+ 1199041199382

||minus 1199041199383

||minus 1199031199384

||


1199291perp = 1198951199381

perp + 1198961199382perp + 1198961199383

perp + 1198951199384perp + 1198941199385

perp = 1199021198991199381perp + 1199021198991199384

perp + 1199011198991199385perp

1199292perp = 1199031199381

perp + 1199041199382perp minus 1199041199383



1199291perp 1199292

perp = 119930 1199291 1199292

119930 = 1199291perp 1199292




72




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79




73




ഥ119938119895 = 119938119895|| 119938119895

perp rarr 119938119895 = 119938119895|| 119939119895

perp


Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Approximant No1

1199291

1199292

1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

1

Examples



74


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 119938120783+ 119938120784minus 119938120785minus 119938120786

1

1

1

2



75

1199291

1199292


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


1199291 = 120784119938120783minus 120785119938120784minus 120785119938120785+ 120784119938120786+ 120784119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

1



76

1199291

1199292


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79


1199291 = 119938120783minus 120786119938120784minus 120786119938120785+ 119938120786+ 120788119938120787

1199292 = 120784119938120783+ 119938120784minus 119938120785minus 120784119938120786

1

2

1

2



77

1199291

1199292

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Approximant No5

1199291 = 120785119938120783minus 120789119938120784minus 120789119938120785+ 120785119938120786+ 120790119938120787

1199292 = 120785119938120783+ 120784119938120784minus 120784119938120785minus 120785119938120786

2

3

2

3

Examples



78

1199291

1199292

Exercises






79

Exercises






79

準周期構造の理論的基礎 Theoretical introduction to...

Documents

Transcript of 準周期構造の理論的基礎 Theoretical introduction to...