The cycle indices of the symmetry groups of platonic solids

#### The cycle indices of the symmetry groups of platonic solids

There are special routines for the computation of the cycle indices of the symmetry groups of platonic solids: There are the 5 platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. All the symmetries of these solids act as permutation groups of the sets of vertices, edges and faces of these solids. The symmetries are either rotations or reflections, and the rotations form a subgroup of index 2 of the group of all all symmetries. The cube and the octahedron have the same symmetries (vertices or faces of the cube can be identified with faces or vertices of the octahedron). The same is true for the dodecahedron and the icosahedron. The cycle indices of these symmetries are given in 3-dimensional form (for vertices, edges and faces 3 different families of indeterminates are used).

The routines for computing the 3-dimensional cycle indices of the symmetry groups (or rotational symmetry groups) of the platonic solids are:

```INT zykelind_tetraeder(a)              OP a;
INT zykelind_tetraeder_extended(a)     OP a;
INT zykelind_cube(a)                   OP a;
INT zykelind_cube_extended(a)          OP a;
INT zykelind_dodecahedron(a)           OP a;
INT zykelind_dodecahedron_extended(a)  OP a;
```

The `_extended` versions are the cycle indices of the groups of all symmetries (this means rotations and reflections). In all these cases `a` is the 3-dimensional cycle index for the action on the sets of vertices, edges and faces.

Extracting a single family of indeterminates leads to:

```INT zykelind_tetraeder_vertices(a)              OP a;
INT zykelind_tetraeder_edges(a)                 OP a;
INT zykelind_tetraeder_faces(a)                 OP a;
INT zykelind_tetraeder_vertices_extended(a)     OP a;
INT zykelind_tetraeder_edges_extended(a)        OP a;
INT zykelind_tetraeder_faces_extended(a)        OP a;
INT zykelind_cube_vertices(a)                   OP a;
INT zykelind_cube_edges(a)                      OP a;
INT zykelind_cube_faces(a)                      OP a;
INT zykelind_cube_vertices_extended(a)          OP a;
INT zykelind_cube_edges_extended(a)             OP a;
INT zykelind_cube_faces_extended(a)             OP a;
INT zykelind_dodecahedron_vertices(a)           OP a;
INT zykelind_dodecahedron_edges(a)              OP a;
INT zykelind_dodecahedron_faces(a)              OP a;
INT zykelind_dodecahedron_vertices_extended(a)  OP a;
INT zykelind_dodecahedron_edges_extended(a)     OP a;
INT zykelind_dodecahedron_faces_extended(a)     OP a;
```

The cycle indices for the actions on the sets of vertices, edges, etc. were computed from the 3-dimensional cycle indices by extracting some families of indeterminates.

harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001

 The cycle indices of the symmetry groups of platonic solids