# symmetries of an octahedron

• September 9th 2008, 06:02 AM
wik_chick88
symmetries of an octahedron
hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces. thankssss
• September 9th 2008, 07:39 AM
ThePerfectHacker
Quote:

Originally Posted by wik_chick88
hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces. thankssss

Label the vertices of the octahedron from $1$ to $6$ in "order". What does it it mean in order? It mean we have $1,2,3,4$ all along the same square with $5,6$ in opposite sides - not sure if that makes sense. Now we need to replace the octahedron in such a way so that "order" is preserved. There are six ways to place the vertex $1$ and then four possible places to choose for $2$. After that is doen the "order" must be contained so along that square $3$ and $4$ are determined. This leaves us with $5,6$ for which there are two locations on opposite sides. Multiply out the numbers and get that the symmetry group has order 48. You can find it here
• September 9th 2008, 08:56 AM
Opalg
Quote:

Originally Posted by wik_chick88
hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces.

It depends what sort of transformations you are allowing. As explained by ThePerfectHacker, the full symmetry group of a regular octahedron has order 48. But only 24 of these can be physically realised by rotations. The other 24 are mirror images of the original octahedron.
• September 9th 2008, 11:05 PM
wik_chick88
ok well i need to figure out how many distinct octahedron i get if i can colour each face either red, blue or green. this is what i worked out. you said that there are 24 PHYSICAL rotations, not including mirror images. there are (8 + 3 - 1) choose 3 = 10 choose 3 = 120 different combinations of 3 colours on the eight faces. does this mean that there are 120/24 = 5 distinct octahedra with coloured faces? that cant be right, its not nearly enough. im very confused...