Polyhedral compound

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Stella_octangula.png
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.

External link

  • Compound polyhedra (http://www.georgehart.com/virtual-polyhedra/compounds-info.html) – from Virtual Reality Polyhedra
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