Cuboctahedron

The cuboctahedron is a polyhedron with eight triangular faces and six square faces. It has a total of 2 edges, each of which borders a triangular face and a square face; and it has 12 vertices, each serving as a corner for two triangle faces and two square faces.

It belongs to the cube and octahedron family of polyhedra, and has the same symmetry. In fact, one can think of the cuboctahedron as a cube with the vertices cut off.

It is interesting to examine how the axes of symmetry correspond exactly between the cube and the octahedron:

• The cube has three axes of four-fold symmetry that pass through the centres of opposite faces. If we treat the cuboctahedron as a cube with the vertices cut off, then these cube face centres are the centres of the cuboctahedron’s square faces. Sure enough, the cuboctahedron has fourfold symmetry around axes that pass through the centres of opposite square faces, and there are three of them.
• The cube has three-fold symmetry around each of four axes that pass through opposite vertices. In the cuboctahedron these vertices have been cut off, leaving triangular faces in their place; so instead the cuboctahedron has three-fold symmetry around each of four axes that pass through the centres of opposite triangular faces.
• The cube has two-fold symmetry around axes that pass through opposite edge centres. The truncation of vertices to form the cuboctahedron reduces each edge to a single point — the cuboctahedron’s vertices. Sure enough, the cuboctahedron has twofold symmetry around vertex axes.
• This exhausts the cube’s rotational symmetries, but we haven’t considered the cuboctahedron’s edge centres yet. However each edge of a cuboctahedron borders two differently shaped faces, so the cuboctahedron is not symmetric about its edge centres.

Rainbow Cube, solved.

The implication for cuboctahedron puzzles is that they must correspond somewhat to certain cube puzzles and octahedron puzzles. A cuboctahedron puzzle that turns about triangular faces, such as the Rainbow Cube, corresponds closely to a vertex-turning cube puzzle (or a face-turning octahedron puzzle) — in fact the Rainbow Cube corresponds exactly to the Dino Cube. A cuboctahedron puzzle that turns about square faces would correspond closely to a face-turning cube puzzle. And a vertex-turning cuboctahedron puzzle would correspond to an edge-turning cube puzzle.