Computing the Symmetry Groups of the Platonic Solids with the Help of Maple

In this note we will determine the symmetry groups of the Platonic solids. We will use Maple to help us do this. The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. We will view the symmetry groups of these solids in two ways. By definition, the symmetry group of a solid is the set of isometries of R that stabilize the set. In each of these five cases we may view the solid as being centered at the origin. The center of each must be fixed by any symmetry. Thus, the symmetry group is then a subgroup of the group of linear isometries, which is the orthogonal group O3(R) of linear transformations that preserve dot products. A second interpretation of the symmetry groups is that the group is a subgroup of the group of permutations of the vertices of the solid. We may view O3(R) as the group of matrices { A ∈ Gl3(R) : AA = I } . Then SO3(R) = {A ∈ O3(R) : det(A) = 1} is a subgroup of index 2, and it coincides with the group of rotations of R. If z : R → R is defined by z(x) = −x for all x ∈ R, then z is a reflection, z is central in O3(R), and O3(R) = SO3(R) × 〈z〉 ∼= SO3(R) × Z2. The element z is a symmetry of all the Platonic solids except for the Tetrahedron. Thus, it will be sufficient for the remaining four solids to determine the rotation subgroup. For each of the Platonic solids, we will use a counting argument to find an upper bound for the order of the symmetry group. We will then, by intelligent trial and error, find three elements of the symmetry group and use Maple to show that the group generated by them has order exactly equal to this upper bound. By doing so, we can conclude that the upper bound is in fact equal to the order of the group, and that the group is generated by the three symmetries. In each case we will have the symmetry group G represented as G = 〈a, b, c〉 with a, b rotations and c a reflection. To help our counting argument, we point out that any symmetry is determined by its action on three vertices; to see this, if we center the solid at the origin, then three vertices represent three linearly independent vectors, or a basis of R. A symmetry fixes the center, so is a linear transformation, which is then determined by its