Truncated octahedron

Truncated octahedron

Dayan Gem, solved

The truncated octahedron has 14 faces, comprising 6 square faces and 8 hexagon faces. It has 36 edges and 24 vertices. Each of its 24 vertices is bordered by a square face and two hexagonal faces, so each vertex has exactly one isometry with each other vertex.

The truncated octahedron is so named because it can be constructed by cutting a square pyramid off each vertex of an octahedron. If you cut a little deeper, to the point where your cuts meet, you would end up with a cuboctahedron. All of these polyhedra — the octahedron, its dual the cube, the cuboctahedron and the truncated octahedron — belong to the same family of polyhedra, and all have the same symmetry: a total of 24 isometries, with 13 axes of rotational symmetry, consisting of three axes of 90° symmetry, 4 of 120° symmetry, and 6 of 180° symmetry. Thus you would expect a truncated octahedron puzzle to make use of the same core as a cube puzzle.

In the truncated octahedron, the three axes of 90° symmetry pass through the centres of opposite square faces; the four axes of 120° symmetry pass through the centres of opposite hexagonal faces; and the size axes of 180° symmetry pass through the centre of opposite edges between hexagonal faces.

There are no axes of rotational symmetry that pass through vertices. Therefore: there can be no such thing as a vertex-turning truncated octahedron puzzle. (Actually that’s not quite true; there can be, but it would only take the shape of a truncated octahedron when solved. Unsolved, it would be jumbled.)