# Thread: symmetries of an octahedron

1. ## 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

2. 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 $\displaystyle 1$ to $\displaystyle 6$ in "order". What does it it mean in order? It mean we have $\displaystyle 1,2,3,4$ all along the same square with $\displaystyle 5,6$ 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 $\displaystyle 1$ and then four possible places to choose for $\displaystyle 2$. After that is doen the "order" must be contained so along that square $\displaystyle 3$ and $\displaystyle 4$ are determined. This leaves us with $\displaystyle 5,6$ 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

3. 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.

4. 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...