Polyhedral compound
From Wikipedia, the free 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.
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[edit] Regular compounds
A regular polyhedron compound can be defined as a compound which is vertex-uniform, edge-uniform, and face-uniform. With this definition there are 5 regular compounds.
| Components | Picture | Vertices | Face-planes | Symmetry |
|---|---|---|---|---|
| Compound of two tetrahedra |  |
Cube | Octahedron | Oh |
| Compound of five tetrahedra |  |
Dodecahedron | Icosahedron | I |
| Compound of ten tetrahedra |  |
Dodecahedron | Icosahedron | Ih |
| Compound of five cubes |  |
Dodecahedron | Rhombic triacontahedron | Ih |
| Compound of five octahedra |  |
Icosidodecahedron | Icosahedron | Ih |
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 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.
[edit] Uniform compounds
In 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds (including infinite prismatic sets of compounds) made from uniform polyhedra with rotational symmetry. (Every vertex is vertex-uniform and every vertex is transitive with every other vertex.) [1]
[edit] Dual-regular compounds
There are four compounds which are composed of a regular polyhedron and its dual.
[edit] External links
- Stella: Polyhedron Navigator - Software which can print nets for many compounds.
- Compound polyhedra – from Virtual Reality Polyhedra
- Skilling's 75 Uniform Compounds of Uniform Polyhedra
- Skilling's Uniform Compounds of Uniform Polyhedra
- Polyhedral Compounds
- http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
- http://www.progonos.com/furuti/Origami/Modular/virtual.html
[edit] References
- John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.
- Peter R. Cromwell, Polyhedra, Cambridge, 1997.
- Michael G. Harman, Polyhedral Compounds, unpublished manuscript, circa 1974. [2]
- Edmund Hess 1876 "Zugleich Gleicheckigen und Gleichflächigen Polyeder", Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg 11 (1876) pp 5-97.
- Luca Pacioli, De Divina Proportione, 1509.
