Coloring A Snub Cube With Restrictions

Question

How many ways can you colour the triangles of a snub cube with black and white?

a snub cube has 32 triangles so i'd assume $2^{32}$ but thats too simple, based on the motions of symmetry of a snub cube, which i know permute the diagonals of a cube.

How many ways are there of coloring the triangular faces of the snub cube with black and white using 16 black faces and 16 white faces.

Given that there are 32 triangles, some of the rotations will give a similar colouring. But i dont know how to go about it.

Answer 1

You will need to use Pólya enumeration, also known as Burnside's lemma. To do this, we need to find the cycle index of the group of rotations of the snub cube. Note that this group is the same as the octahedral group, with order $24$.

• The identity, which maps all $6(4)+8 = 32$ triangles to themselves: $a_1^{32}$.
• Six rotations by $\pi/2$ about an axis through the centers of opposite square faces. Because we are looking at how such a rotation affects the triangles, clearly there is no triangle that remains fixed (unlike the square faces, in which two map to themselves). Every triangular face has order $4$ (exactly four such rotations will return each triangle to itself), so the contribution to the cycle index is $6a_4^{8}$; that is to say, there are $8$ four-cycles.
• Three rotations by $\pi$ about an axis through the centers of opposite square faces. These create $16$ two-cycles, so the contribution is $3a_2^{16}$.
• Eight rotations by $2\pi/3$ about an axis through the centers of opposite non-snub triangular faces (i.e. through triangles not edge-incident with a square face). Such a rotation will fix the two triangles on the axis of rotation, and all of the others will belong to three-cycles; thus the contribution is $8a_1^2 a_3^{10}$.
• Six rotations by $\pi$ through the opposing midpoints of the edge joining two snub triangles. Here, there are no fixed triangles, and each triangle belongs to one of $16$ two-cycles, and the contribution to the cycle index is $6a_2^{16}$.

The total cycle index is therefore $$|Z(a_1, a_2, a_3, a_4)| = \frac{1}{24} \left(a_1^{32} + 6a_4^8 + 3a_2^{16} + 8a_1^2 a_3^{10} + 6a_2^{16}\right).$$

Therefore, with $m$ colors, there are $$|Z(m,m,m,m)| = \frac{1}{24}(m^{32} + 9m^{16} + 8m^{12} + 6m^8)$$ colorings. For $m = 2$, this gives $2^6 \cdot 3 \cdot 932203 = 178982976$ colorings.