Symmetry groups of the platonic solids Mathematics, Statistics and Computer Science Department...
-
Upload
jessica-diane-jackson -
Category
Documents
-
view
219 -
download
0
Transcript of Symmetry groups of the platonic solids Mathematics, Statistics and Computer Science Department...
Symmetry groups of the platonic solids Mathematics, Statistics and Computer Science Department Xiaoying (Jennifer) Deng
Outline 0Introduction0Properties0Rotational symmetry groups of some platonic solids0Related groups0Future work0Exam question
Introduction0Definition: A platonic solid is a convex polyhedron that
is made up of congruent regular polygons with the same number of faces meeting at each vertex.
✗✗
OctahedronHexahedron (Cube)
Tetrahedron
IcosahedronDodecahedron
Euler’s formula
Name F E V
Tetrahedron 4 6 4
Cube 6 12 8
Octahedron 8 12 6
Dodecahedron 12 30 20
Icosahedron 20 30 12
F + V - E = 2
Duality0Definition: A dual of a polyhedron is formed by 0place points on the center of every faces0 connect the points in the neighbouring faces of the original
polyhedron to obtain the dual
Cube IcosahedronTetrahedron
Lemma: Dual polyhedra have the same symmetry groups.
Symmetry group0Definition:0Let X be a platonic solid.0Rotational(Direct) symmetry group of X is a symmetry
group of X if only rotation is allowed.0Full symmetry group of X is a symmetry group of X if
both rotation and reflection are allowed.
0For a finite set A of n elements, the group of all permutations of A is the symmetric group on n letters.
The Tetrahedron
Rotational symmetry Permutations of 4 numbers
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
01200 ; Two new symmetries for each vertex.04 × 2 = 8 new symmetries. 0Vertex 1; (2, 4, 3) (2, 3, 4)0Vertex 2; (1,3, 4) (1, 4, 3)0Vertex 3; (1, 2, 4) (1, 4, 2)0Vertex 4; (1, 2, 3) (1, 3, 2)
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
01800 ; One symmetry for each axis.03 × 1 = 3 new symmetries.0 (1, 2)(3, 4) (1, 3)(1, 4) (1, 4)(2, 3)
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
0 (1, 2)(1, 2)01 + 8 + 4 = 12 rotational symmetries.
0The alternating group: A4
https://www.youtube.com/watch?v=qAR8BFMS3Bc ( 2:01)
The cube
http://www.youtube.com/watch?v=gBg4-lJ19Gg (1:38)
0 1200 ; Two new symmetries for each axis.0 4 × 2 = 8 new symmetries.0 1800 ; One symmetry for each axis.0 6 × 1 = 6 new symmetries.
0 1 + 9 + 8 + 6 = 24 rotational symmetries.
0 S4
0 900 ; Three new symmetries for each axis.0 3 × 3 = 9 symmetries
d1
d4
d3
d2
The Octahedron
Name Rotationalsymmetries
Rotation Group Dual
Tetrahedron 12 A4 Tetrahedron
Cube 24 S4 Octahedron
Octahedron 24 S4 Cube
Future work 0Reflection group of platonic
solids
0Reflection group of the tetrahedron
0Full symmetry group of the tetrahedron
0The rotational symmetry group of the dodecahedron and the Icosahedron
Name Rotational symmetries
Rotation Group Dual
Dodecahedron 60 A5 Icosahedron
Icosahedron 60 A5 Dodecahedron
Name Orbit(vertices)
Stabilizer (faces at each vertex)
|G+|
Tetrahedron 4 3 12
Cube 8 3 24
Octahedron 6 4 24
Dodecahedron 20 3 60
Icosahedron 12 5 60
0Stabilizer ; The Orbit-Stabilizer Theorem
Exam QuestionHow many rotational symmetries of the cube?
0900 ; 3 × 3 = 9 symmetries.01200 ; 4 × 2 = 8 new symmetries.01800 ; 6 × 1 = 6 new symmetries.01 + 9 + 8 + 6 = 24 rotational symmetries.
Solution:
Reference 0 Kappraff, J. (2001). Connections: The geometric bridge between art and
science. Singapore: World Scientific.0 Hilton, P., Pedersen, J., & Donmoyer, S. (2010). A mathematical tapestry:
Demonstrating the beautiful unity of mathematics. New York: Cambridge University Press.
0 Senechal, M., Fleck, G. M., & Sherer, S. (2012). Shaping space: Exploring polyhedra in nature, art, and the geometrical imagination. New York: Springer.
0 Berlinghoff, W. P., & Gouvêa, F. Q. (2004). Math through the ages: A gentle history for teachers and others. Washington, DC: Mathematical Association of America.
0 Richeson, D. S. (2008). Euler's gem: The polyhedron formula and the birth of topology. Princeton, N.J: Princeton University Press.
0 In Celletti, A., In Locatelli, U., In Ruggeri, T., & In Strickland, E. (2014). Mathematical models and methods for planet Earth.
?