Quote:
Originally Posted by wik_chick88  how many distinct octahedra (all of the same size) are possible if each face of each octahedron is either red, blue or green? the octahedron is a regular solid with 6 vertices, 12 edges and 8 triangular faces.
im having trouble working out how many symmetries there are of an octahedron. 8 faces and 3 colours so different ways to colour the faces would be 8 choose 3 = 56. but obviously 2 ways might have the same physical result if the octahedron was rotated on a vertice, edge or face. |
This is a way to approach this problem.
Let
be the group of all trasformations of the octahedron. In the other thread it was shown that
. And
set of all colorings (even the possibly similar one).
Now put these symettries into conjugacy classes which can be found
here. For each conjugacy class, say 90 degree rotation, find which elements in
are fixed and how many of them. This number gets multipled by six since there are 90 degree rotations. And go through each class. After you done the hard part the rest follows by
Burnside's lemma - be sure to look at the example with the cube in that link, it is very similar.