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Benoit Mandelbrot

- a mathematician

Fractals:

Geometrical objects with

More & more detail

on smaller scales

Self-similarity

(symmetry under scaling

or magnification)

Fractional dimensions

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Power laws with long tails

y =1

x!

! > 0( )

Hyperbola: y =

1

x

2 4 6 8 10x

0.2

0.4

0.6

0.8

1y

Bell curve: y = e!x2

2 4 6 8 10x

0.2

0.4

0.6

0.8

1y

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Separation of scales:

a basis for reductionism

A hierarchy of scales

Microscopic scales ~10!10

m

Macroscopic scales ~ 1 m

-> Usually, no need to worryabout the microscopic level

when we try to understand

macroscopic phenomena

Fractals: no separation of scales

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Symmetry under scaling

or resizingor magnification

Hyperbola

2 4 6 8 10x

0.2

0.4

0.6

0.8

1y

0.2 0.4 0.6 0.8 1x

2

4

6

8

10y

A Gaussian

2 4 6 8 10x

0.2

0.4

0.6

0.8

1y

0.2 0.4 0.6 0.8 1x

0.2

0.4

0.6

0.8

1y

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Zips law

on the English language

A power law (a hyperbola):

Frequency( ) =Constant( )

Ranking( )

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The Cantor setStage Length of

segmentNumber ofsegments

Total length =(# of seg.) ! (length of seg.)

0 1 1 1

1 1/3 2 2 ! (1/ 3)= 2 / 3 = 0.666

2 (1 / 3) ! (1 / 3)

=(1 / 3)2= 1 / 9

2 ! 2 = 22

= 4 4 ! (1 / 9) = (2/ 3)2

= 0.444

3 (1 / 3)3= 1/ 27 2 ! 2

2= 2

3= 8 8 ! (1 / 27) = (2/ 3)

3=0.296

4 (1 / 3)4= 1 / 81 2 ! 2

3= 2

4= 16

2

4! (1 / 3)

4= (2 / 3)

4=0.198

5 (1 / 3)5= 1 / 243 2 ! 2

4= 2

5= 32

25! (1 / 3)

5= (2 / 3)

5=0.132

n (1 / 3)n

2n

2n

! (1 / 3)n

= (2 / 3)n

! 0 ! 0

5 10 15 20 n

0.2

0.4

0.6

0.8

1

Length

At an infinite# of stages:

an infinite# of segments

of an infinitesimallength

but the total length = 0

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Hidden power law

Count the # of gaps for each siz

1 gap of size 1 3 = 0.333

2 gaps of size 1 3( )2

= 0.111

4 gaps of size 1 3( )3

= 0.037

# of gaps with size s( ) =1

2

1

s0.6309

Magnifyxby 3 &yby 2

-> scaling

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The Koch curve

Stage Length of

segment

Number of

segments

Total length or Perimeter =

(# of seg.) ! (length of seg.)

0 1 3 3

1 1/3 3! 4 = 12 12 ! (1 / 3)=4

2 (1 / 3) ! (1 / 3)

= (1/3)2= 1/9

3! 4 ! 4

= 3! 42= 48

48 ! (1/ 9) = 3 ! (4 / 3)2

=16 / 3 = 5.33

3 (1/3)3= 1/27 3! 43= 192 192 ! (1 / 27) = 3 ! (4 / 3)

3

= 64 / 9 = 7.11

4 (1 / 3)4 3! 4

4= 768 3 ! (4 / 3)

4

= 256 / 27 = 9.48

5 (1 / 3)5 3! 4

5= 3072 3 ! (4 / 3)

5

= 1024 / 81 =12.64

n (1/3)n= 1/3n 3! 4n

3 ! (4 / 3)n

=3 ! (1.33)n

! 0 ! !

At an infinite# of stages:

an infinite# of segments

of an infinitesimallength

but the infinitetotal length

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The Sierpinski carpet

Stage Area ofsubsquares

Number ofsubsquares

Total Area =(# of sq.) ! (area of sq.)

0 1 1 1

1 1/9 9 - 1= 8 8/9 = 0.89

2 (1 / 9) ! (1 / 9)

=(1 / 9)2=1 / 81

8! 8= 82= 64 64 / 81= (8 / 9)

2=0.79

3 (1 / 9)3= 1 / 729 8

3= 512 512 / 729 =(8 / 9)

3=0.70

4 (1 / 9)4= 1/ 6561

84= 4096 4096 / 6561=(8/ 9)

4=0.62

5 (1 / 9 )5 8

5 (8 / 9)

5= 0.55

n (1 / 9 )n

8n

(8 / 9)n

= (0.89)n

! 0 ! 0

At an infinite# of stages:

an infinite# of subsquares

of an infinitesimalareabut the total area = 0

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The box-counting dimension

A line segment

Scale factor

x

1/x Number of

boxes,N

1 1 1

1/2 2 2

1/4 4 4

1/8 8 8

1/n n n

# of boxes( ) =

1

Scale factor x( )

"# %&

1

' D = 1

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A square

Scale factor

x

1/x Number of

boxes,N

1 1 1

1/2 2 4

1/4 4 16

1/8 8 64

1/n n n2

# of boxes( ) =1

Scale factor x( )

"#

%&

2

'D

=2

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On a double logarithmic graph

Plot # of boxes vs. 1 x

Fit the data points w/ a line

D ="Rise"

"Run"

Run

RiseN

1/x 1

10

100

1000

1 10 100

N(line)N(square)

N

1/x

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The Cantor set

Scale factorx

1/x Number ofboxes,N

1 1 1

1/3 3 2

1 / 3( )2 32= 9 2

2= 4

1 / 3( )3 33= 27 2

3= 8

1 / 3( )4 3

4= 81 2

4= 16

(1 / 3)n

3n

2n

1

10

100

1000

1 10 100

N (Cantor)N (line)N (square)

N

1/x

D =

ln2

ln3=

0.6931...

1.0986...=0.6309...

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The Sierpinski carpetScale factor

x

1/x Number ofboxes,N

1 1 1

1/3 3 8

1 / 9 32= 9 8

2= 64

1/27 33= 27 8

3= 512

1/81 34= 81 8

4= 4096

1 / 3n

3n

8n

1

10

100

1000

1 10 100

N (Sierpinski carpet)N (line)N (square)

N

1/x

D = ln8ln3

= 2.0794...1.0986...

=1.8928...