Polyhedral compound

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

Neighbouring 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 considered the core for a set of stellations including this compound. (See List of Wenninger polyhedron models for these compounds and more stellations.)

Contents

Regular compounds

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.

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.

Dual-regular compounds

A dual-regular compound is composed of a regular polyhedron (one of the Platonic solids or Kepler-Poinsot polyhedra) and its regular 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.

The dual-regular compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula, since the tetrahedron is self-dual.

The cube-octahedron and dodecahedron-icosahedron dual-regular compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.

The compound of the small stellated dodecahedron and great dodecahedron looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside. For this reason, the image shown above shows the small stellated dodecahedron in wireframe.

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