Polyhedral compound

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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, 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 stellation thereof.

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

Regular compounds

Components Picture Vertices Face-planes Symmetry
2 tetrahedra 100px Cube Octahedron Oh
5 tetrahedra 100px Dodecahedron Icosahedron I
10 tetrahedra 100px Dodecahedron Icosahedron Ih
5 cubes 100px Dodecahedron Rhombic triacontahedron Ih
5 octahedra 100px 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.

Dual-regular compounds

There are four compounds which are composed of a regular polyhedron and its dual.

Components Picture Symmetry
2 tetrahedra 100px Oh
Cube and octahedron 100px Oh
Dodecahedron and
100px Ih
Great icosahedron and
Great stellated dodecahedron
100px Ih

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