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, 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 | O_{h} |
5 tetrahedra | 100px | Dodecahedron | Icosahedron | I |
10 tetrahedra | 100px | Dodecahedron | Icosahedron | I_{h} |
5 cubes | 100px | Dodecahedron | Rhombic triacontahedron | I_{h} |
5 octahedra | 100px | Icosidodecahedron | Icosahedron | I_{h} |
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 | O_{h} |
Cube and octahedron | 100px | O_{h} |
Dodecahedron and icosahedron |
100px | I_{h} |
Great icosahedron and Great stellated dodecahedron |
100px | I_{h} |