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.
| Components | Picture | Convex hull | Core | Symmetry | Subgroup restricting to one constituent |
Dual |
|---|---|---|---|---|---|---|
| Compound of two tetrahedra, or stella octangula | Cube | Octahedron | *432 Oh |
*332 Td |
Self-dual | |
| Compound of five tetrahedra, or chiro-icosahedron | Dodecahedron | Icosahedron | 532 + I |
332 + T |
enantiomorph, or chiral twin | |
| Compound of ten tetrahedra, compound of two chiro-icoshedra, or icosiicosahedron | Dodecahedron | Icosahedron | *532 Ih |
332 T |
Self-dual | |
| Compound of five cubes, or rhombihedron | Dodecahedron | Rhombic triacontahedron | *532 Ih |
3*2 Th |
Compound of five octahedra | |
| Compound of five octahedra, or small icosiicosahedron | Icosidodecahedron | Icosahedron | *532 Ih |
3*2 Th |
Compound of five cubes |
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.
Read more about this topic: Polyhedral Compound
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