Polyhedral compound

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

The outer vertices of a compound can be connected to form a convex polyhedron called the convex hull. The compound is a facetting of the convex hull.

Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations.

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. There are five regular compounds of polyhedra.

Best known is the compound of two tetrahedra, often called the stella octangula, a name given to it by Kepler. The vertices of the two tetrahedra define a cube and the intersection of the two an octahedron, which shares the same face-planes as the compound. Thus it is a stellation of the octahedron, and in fact, the only finite stellation thereof.

The stella octangula can also be regarded as a dual-regular compound.

The compound of five tetrahedra comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra. Each of the tetrahedral compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra.

A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds of the regular polyhedra.

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