Polytope compound

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

The outer 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 used as the core for a set of stellations.

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. There are five regular compounds of polyhedra.

Components Coxeter symbol	Picture	Spherical	Convex hull (Core)	Symmetry	Subgroup restricting to one constituent	Dual
Two tetrahedra {4,3}[2{3,3}]{3,4}			Cube (Octahedron)	*432 [4,3] Oh	*332 [3,3] Td	Self-dual
Five tetrahedra {5,3}[5{3,3}]{3,5}			Dodecahedron (Icosahedron)	532 [5,3]+ I	332 [3,3]+ T	enantiomorph chiral twin
Ten tetrahedra 2{5,3}[10{3,3}]2{3,5}			Dodecahedron (Icosahedron)	*532 [5,3] Ih	332 [3,3] T	Self-dual
Five cubes 2{5,3}[5{4,3}]			Dodecahedron (Rhombic triacontahedron)	*532 [5,3] Ih	3*2 [3,3] Th	Five octahedra
Five octahedra [5{3,4}]2{3,5}			Icosidodecahedron (Icosahedron)	*532 [5,3] Ih	3*2 [3,3] Th	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 compounds

A dual compound is composed of a polyhedron and its 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 of the regular polyhedra.

Components	Picture	Convex hull	Core	Symmetry
two tetrahedra (stella octangula)		Cube	Octahedron	*432 [4,3] Oh
cube and octahedron		Rhombic dodecahedron	Cuboctahedron	*432 [4,3] Oh
dodecahedron and icosahedron		Rhombic triacontahedron	Icosidodecahedron	*532 [5,3] Ih
great icosahedron and great stellated dodecahedron		Dodecahedron	Icosidodecahedron	*532 [5,3] Ih
small stellated dodecahedron and great dodecahedron		Icosahedron	Dodecahedron	*532 [5,3] Ih

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

The cube-octahedron and dodecahedron-icosahedron dual 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. For this reason, the image shown above shows the small stellated dodecahedron in wireframe.

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-transitive and every vertex is transitive with every other vertex.) This list includes the five regular compounds above.

The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element. Some chiral pairs of face groups 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

Other compounds

These compounds, of four cubes, and (dual) four octahedra, are neither regular compounds, nor dual compounds, nor uniform compounds.

• Compound of three octahedra
• Compound of four cubes

Two polyhedra that are compounds but have their elements rigidly locked into place are the small complex icosidodecahedron (compound of icosahedron and great dodecahedron) and the great complex icosidodecahedron (compound of small stellated dodecahedron and great icosahedron). If the definition of a uniform polyhedron is generalised they are uniform.

The section for entianomorphic pairs in Skilling's list does not contain the compound of two great snub dodecicosidodecahedra, as the pentagram faces would coincide. Removing the coincident faces results in the compound of twenty octahedra.

4-polytope compounds

Orthogonal projections

75 {4,3,3}	75 {3,3,4}

In 4-dimensions, there are a large number of regular compounds of regular polytopes. Coxeter lists a few of them in his book Regular Polytopes:^[1]

Self-duals:

Dual pairs:

Uniform compounds and duals with convex 4-polytopes:

Dual positions:

Compounds with regular star 4-polytopes

Self-dual star compounds:

Dual pairs of compound stars:

Uniform compound stars and duals:

Group theory

In terms of group theory, if G is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if H is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to.

Compounds of tilings

There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated.

The Euclidean and hyperbolic compound families 2 {p,p} (4 ≤ p ≤ ∞, p an integer) are analogous to the spherical stella octangula, 2 {3,3}.

A few examples of Euclidean and hyperbolic regular compounds

Self-dual	Duals		Self-dual
2 {4,4}	2 {6,3}	2 {3,6}	2 {∞,∞}
	3 {6,3}	3 {3,6}	3 {∞,∞}

A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs.

There are also dual-regular tiling compounds. A simple example is the E² compound of a hexagonal tiling and its dual triangular tiling. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.

Footnotes

External links

• MathWorld: Polyhedron Compound
• 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
• Compound of Small Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2}
• Klitzing, Richard. "Compound polytopes". 

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554 .
• Cromwell, Peter R. (1997), Polyhedra, Cambridge .
• Wenninger, Magnus (1983), Dual Models, Cambridge, England: Cambridge University Press, pp. 51–53 .
• Harman, Michael G. (1974), Polyhedral Compounds, unpublished manuscript .
• Hess, Edmund (1876), "Zugleich Gleicheckigen und Gleichflächigen Polyeder", Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg, 11: 5–97 .
• Pacioli, Luca (1509), De Divina Proportione .
• Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  p. 87 Five regular compounds