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# Polyhedral compound

Stella octangula, a polyhedral compound

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

The best known is the compound of two tetrahedra called the stella octangula , discovered 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 stellation thereof.

The stella octangula is one of only five compounds that are vertex-, edge-, and face-uniform, called regular compounds:

Components Vertices Face-planes Symmetry
2 tetrahedra Cube Octahedron Oh
5 tetrahedra Dodecahedron Icosahedron I
10 tetrahedra Dodecahedron Icosahedron Ih
5 cubes Dodecahedron Rhombic triacontahedron Ih
5 octahedra Icosidodecahedron Icosahedron Ih

The compound of 5 tetrahedra actually 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.

The stella octangula can also be regarded as a compound of a tetrahedron with its dual polyhedron, inscribed in a common sphere so that the vertices of one line up with the face centres of the other. The corresponding cube-octahedron and dodecahedron-icosahedron compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.