Symmetry groups of the platonic solids Mathematics, Statistics and Computer Science Department Xiaoying (Jennifer) Deng

Outline 0 Introduction 0 Properties 0 Rotational symmetry groups of some platonic solids 0 Related groups 0 Future work 0 Exam question

Introduction 0 Definition: 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. ✗ ✗

Octahedron Hexahedron (Cube) Tetrahedron Icosahedron Dodecahedron

Euler’s formula NameFEV Tetrahedron464 Cube Octahedron8126 Dodecahedron Icosahedron F + V - E = 2

Duality 0 Definition: A dual of a polyhedron is formed by 0 place points on the center of every faces 0 connect the points in the neighbouring faces of the original polyhedron to obtain the dual

CubeIcosahedron Tetrahedron Lemma: Dual polyhedra have the same symmetry groups.

Symmetry group 0 Definition: 0 Let X be a platonic solid. 0 Rotational(Direct) symmetry group of X is a symmetry group of X if only rotation is allowed. 0 Full symmetry group of X is a symmetry group of X if both rotation and reflection are allowed. 0 For 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) = 2 P(3) = 3 P(4) = 4

; Two new symmetries for each vertex. 0 4 × 2 = 8 new symmetries. 0 Vertex 1; (2, 4, 3) (2, 3, 4) 0 Vertex 2; (1,3, 4) (1, 4, 3) 0 Vertex 3; (1, 2, 4) (1, 4, 2) 0 Vertex 4; (1, 2, 3) (1, 3, 2) P(1) = 1 P(2) = 2 P(3) = 3 P(4) = 4

; One symmetry for each axis. 0 3 × 1 = 3 new symmetries. 0 (1, 2)(3, 4) (1, 3)(1, 4) (1, 4)(2, 3) P(1) = 1 P(2) = 2 P(3) = 3 P(4) = 4 0 (1, 2)(1, 2) = 12 rotational symmetries. 0 The alternating group: A 4 （ 2:01 ）

The cube (1:38)

; Two new symmetries for each axis. 0 4 × 2 = 8 new symmetries ; One symmetry for each axis. 0 6 × 1 = 6 new symmetries = 24 rotational symmetries. 0 S ; Three new symmetries for each axis. 0 3 × 3 = 9 symmetries d1d1 d4d4 d3d3 d2d2

The Octahedron NameRotational symmetries Rotation GroupDual Tetrahedron12A4A4 Tetrahedron Cube24S4S4 Octahedron 24S4S4 Cube

Future work 0 Reflection group of platonic solids 0 Reflection group of the tetrahedron 0 Full symmetry group of the tetrahedron

0 The rotational symmetry group of the dodecahedron and the Icosahedron NameRotational symmetries Rotation GroupDual Dodecahedron60A5A5 Icosahedron 60A5A5 Dodecahedron

NameOrbit (vertices) Stabilizer (faces at each vertex) |G+| Tetrahedron4312 Cube8324 Octahedron6424 Dodecahedron20360 Icosahedron Stabilizer ; The Orbit-Stabilizer Theorem

Exam Question How many rotational symmetries of the cube?

; 3 × 3 = 9 symmetries ; 4 × 2 = 8 new symmetries ; 6 × 1 = 6 new symmetries = 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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