Polyhedral compound
From Free net encyclopedia
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 (Stella octangula) | Image:Compound of two tetrahedra.png | Cube | Octahedron | Oh |
| 5 tetrahedra (Compound of five tetrahedra) | Image:Compound of five tetrahedra.png | Dodecahedron | Icosahedron | I |
| 10 tetrahedra | Image:Compound of ten tetrahedra.png | Dodecahedron | Icosahedron | Ih |
| 5 cubes | Image:Compound of five cubes.png | Dodecahedron | Rhombic triacontahedron | Ih |
| 5 octahedra | Image:Compound of five octahedra.png | 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 | Image:Compound of two tetrahedra.png | Oh |
| Cube and octahedron | Image:Compound of cube and octahedron.png | Oh |
| Dodecahedron and icosahedron | Image:Compound of dodecahedron and icosahedron.png | Ih |
| Great icosahedron and Great stellated dodecahedron | Image:Compound of great icosahedron and stellated dodecahedron.png | Ih |
External links
- Stella: Polyhedron Navigator - Software which can print nets for many compounds.
- Compound polyhedra – from Virtual Reality Polyhedra
- Polyhedral Compounds
- http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
- http://www.progonos.com/furuti/Origami/Modular/virtual.htmlja:複合多面体