Symmetry Groups of Platonic Solids

Project #7 - From Svetlana Katok's Math 311M Course, Fall 2009

A regular solid is a 3-dimensional polyhedron in which each face is a regular polygon. Any two faces as well as any two vertices can be matched by an isometry (a rigid motion) of the 3-dimensional space. It is convenient to use a symbol {p, q} for a regular solid whose faces are regular p-gons with q of them situated around each vertex.

We have proved that there are five regular solids:

Any regular solid may be inscribed in a sphere, and then any symmetry of any regular solid will leave the center of the sphere fixed and will transform the surface of the sphere onto itself. We call a rotational symmetry of a regular solid any rotation of the sphere (with respect to an axis passing through its center) mapping the regular solid into inself. A symmetry of a regular solid is either a rotational symmetry or a reflection with respect to a plane (also passing through the center of the sphere), mapping the regular solid onto itself.

Prove that:

  1. The group of rotational symmetries of a tetrahedron is A(4)
  2. The full group of symmetries of a tetrahedron is S(4)
  3. The group of rotational symmetries of a cube or of an octahedron is S(4)
  4. The group of rotational symmetries of an icosahedron or of a dodecahedron is A(5)

My Work on the Subject

Symmetry Table

Polyhedron Schläfli
Symbol 		Number of
Faces 		Number of
Vertices 		Number of
Edges 		Rotational
Symmetries 		Full
Symmetries
Tetrahedron {3, 3} 4 4 6 12⇒A(4) 24⇒S(4)
Cube {4, 3} 6 8 12 24⇒S(4) 48⇒S(4)*2
Octahedron {3, 4} 8 6 12
Dodecahedron {5, 3} 12 20 30 60⇒A(5) 12⇒A(5)*2
Icosahedron {3, 5} 20 12 30

Discussion

The order of the rotational symmetries is easily obtained by any of several methods:

The full number of symmetries is twice the number of rotational symmetries as each possible rotation has a unique reflection across any suitable plane.

Once the order of the symmetry group is determined, it still remains necessary to prove the isomorphism of the symmetry group to the permutation groups, which can be done by defining the construction of the polyhedra in terms of "building blocks" whose positions can be permuted to represent the symmetries of the polyhedra. Not all permutations will correspond to a symmetry, so these limitations must also be addressed.

The Symmetries