Cuboctahedron - Geometric Relations

Geometric Relations

A cuboctahedron can be obtained by taking an appropriate cross section of a four-dimensional 16-cell.

A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.

The cuboctahedron is a rectified cube and also a rectified octahedron.

It is also a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2.

A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group.

The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola, also called an anticuboctahedron.

Both triangular bicupolae are important in sphere packing. The distance from the solid's center to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center.

Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform polychora.

The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.