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.

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