Octahedron rotational symmetries help

Question

So consider the symmetry group G of the regular octahedron i.e. an octahedron whose sides are all the same size and shape.

There are 13 axes of symmetry but now I want to determine all the elements of the group of rotational symmetries G.

A hint was to consider how rotational symmetries act to permute the six points of an octahedron.

Can someone help me with this?

Answer 1

Well, one way to solve this is to consider Euler's formula and the number of Vertices, Edges, and Faces for the Octahedron (equivalently we could look at the dual solid, the CUBE).

 

We have 6 vertices, 12 edges, and 8 faces (V+F=E+2) in the octahedron (if we flip faces and vertices we have the dual solid).

There are a few different ways to rotate the solid...one is to rotate along the axis created by pairs of opposite faces...another is from pairs of opposite edges and the other is pairs of opposite vertices...

 

Next we count the number of rotations possible for each axis and multiply by the number of possible pairs (axes of rotation), to get all of the possible rotations for the solid.

 

For opposite edges we have only 1 rotation (by \pi), for opposite faces we can rotate by +/-(2\pi)/3, and for opposite vertices we can rotate by multiples of \pi/2...giving us a total of 24 rotations.

 

If we now number the vertices (as was hinted) and writing out each of these 24 permutations, we can check for isomorphism to other well known symmetric groups!