Polytope compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram.

Neighbouring vertices of a compound can be connected to form a convex polyhedron called the convex hull. The compound is a facetting of the convex hull.

Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be considered the core for a set of stellations including this compound. (See List of Wenninger polyhedron models for these compounds and more stellations.)

Regular compounds

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-uniform, edge-uniform, and face-uniform. With this definition there are 5 regular compounds.

Components	Picture	Convex hull	Core	Symmetry	Dual
Compound of two tetrahedra, or Stella octangula		Cube	Octahedron	Oh	Self-dual
Compound of five tetrahedra		Dodecahedron	Icosahedron	I	enantiomorph, or chiral twin
Compound of ten tetrahedra		Dodecahedron	Icosahedron	Ih	Self-dual
Compound of five cubes		Dodecahedron	Rhombic triacontahedron	Ih	Compound of five octahedra
Compound of five octahedra		Icosidodecahedron	Icosahedron	Ih	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.

Dual-regular compounds

A dual-regular compound is composed of a regular polyhedron (one of the Platonic solids or Kepler-Poinsot solids) and its regular dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds.

Components	Picture	Convex hull	Core	Symmetry
Compound of two tetrahedra		Cube	Octahedron	Oh
Compound of cube and octahedron		Rhombic dodecahedron	Cuboctahedron	Oh
Compound of dodecahedron and icosahedron		Rhombic triacontahedron	Icosidodecahedron	Ih
Compound of great icosahedron and great stellated dodecahedron		Dodecahedron	Icosahedron	Ih
Compound of small stellated dodecahedron and great dodecahedron		Icosahedron	Dodecahedron	Ih

The dual-regular compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula, since the tetrahedron is self-dual.

The cube-octahedron and dodecahedron-icosahedron dual-regular compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.

The compound of the small stellated dodecahedron and great dodecahedron looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside.

Uniform compounds

In 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds (including 6 as infinite prismatic sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is vertex-uniform and every vertex is transitive with every other vertex.) This list includes the five regular compounds above. [1]

Here is a pictorial table of the 75 uniform compounds as listed by Skilling. Most are singularly colored by each polyhedron element. Some as chiral pairs are colored by symmetry of the faces within each polyhedron.

• 1-19: Miscellaneous (4,5,6,9,17 are the 5 regular compounds)

• 20-25: Prism symmetry embedded in prism symmetry,

• 26-45: Prism symmetry embedded in octahedral or icosahedral symmetry,

• 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,

• 68-75: enantiomorph pairs

External links

• MathWorld: Polyhedron Compound
• Stella: Polyhedron Navigator - Software which can print nets for many compounds.
• Compound polyhedra – from Virtual Reality Polyhedra
  • Uniform Compounds of Uniform 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
• Compound of Small Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2}

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.
• Magnus Wenninger Dual Models Cambridge, England, Cambridge University Press, 1983. (51-53)
• 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.