[okgkwyvp] automorphic rotations of a cube

6 = (6/2)*2 90-degree rotations through a face

3 = 6/2 180-degree rotations through a face

8 = (8/2)*2 120 degree-rotations through a vertex

6 = 12/2 180-degree rotations through an edge

1 identity (no rotation)

the divisions by 2 are to avoid double counting: each face, vertex, and edge pairs with a corresponding one diametrically opposite.  multiplications by 2 are because you can rotate left or right.  sum is 24.  it is nice that all the rotations can be enumerated this way without overlaps.

it can also be computed as (6 faces)*(4 rotations) = (8 vertices)*(3 rotations) = (12 edges)*(2 rotations) = 24.  these can be interpreted as bringing a face (or vertex or edge) to a canonical location, then doing a rotation.

can you visualize a given sequence of rotations?

previously, turning half of a cube.

the octahedron can do the same set of rotations, swapping face and vertex: octahedral group.

a similar enumeration works for the dodecahrdron ((12 + 12) + 20 + 15 + 1 = 60) and icosahedron (same sum to 60).