Polyhedral compound
A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the threedimensional 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.)
Contents
Regular compounds
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertextransitive, edgetransitive, and facetransitive. 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 [4,3] O_{h} 
*332 [3,3] T_{d} 
Selfdual  
Compound of five tetrahedra, or chiroicosahedron  Dodecahedron  Icosahedron  532 [5,3]^{+} I 
332 [3,3]^{+} T 
enantiomorph, or chiral twin  
Compound of ten tetrahedra, compound of two chiroicoshedra, or icosiicosahedron  Dodecahedron  Icosahedron  *532 [5,3] I_{h} 
332 [3,3] T 
Selfdual  
Compound of five cubes, or rhombihedron  Dodecahedron  Rhombic triacontahedron  *532 [5,3] I_{h} 
3*2 [3,3] T_{h} 
Compound of five octahedra  
Compound of five octahedra, or small icosiicosahedron  Icosidodecahedron  Icosahedron  *532 [5,3] I_{h} 
3*2 [3,3] T_{h} 
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 faceplanes 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 dualregular 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 selfdual, and the compound of 5 cubes is dual to the compound of 5 octahedra.
Dualregular compounds
A dualregular compound is composed of a regular polyhedron (one of the Platonic solids or KeplerPoinsot polyhedra) 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, or Stella octangula  Cube  Octahedron  *432 [4,3] O_{h} 

Compound of cube and octahedron  Rhombic dodecahedron  Cuboctahedron  *432 [4,3] O_{h} 

Compound of dodecahedron and icosahedron  Rhombic triacontahedron  Icosidodecahedron  *532 [5,3] I_{h} 

Compound of great icosahedron and great stellated dodecahedron  Dodecahedron  Icosidodecahedron  *532 [5,3] I_{h} 

Compound of small stellated dodecahedron and great dodecahedron  Icosahedron  Dodecahedron  *532 [5,3] I_{h} 
The dualregular compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula, since the tetrahedron is selfdual.
The cubeoctahedron and dodecahedronicosahedron dualregular 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 vertextransitive 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.
 119: Miscellaneous (4,5,6,9,17 are the 5 regular compounds)
 2025: Prism symmetry embedded in prism symmetry,
 2645: Prism symmetry embedded in octahedral or icosahedral symmetry,
 4667: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
 6875: enantiomorph pairs
Other compounds
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.
Regular polytope compounds
In 4dimensions, there are a large number of regular compounds of regular polytopes. Coxeter lists a few of them in his book Regular Polytopes:
Selfduals:
Compound  Symmetry 

120 5cell  [5,3,3], order 14400 
5 24cell  [5,3,3], order 14400 
Dual pairs:
Compound 1  Compound 2  Symmetry 

3 16cells^{1}  3 tesseracts  [3,4,3], order 1152 
15 16cells  15 tesseracts  [5,3,3], order 14400 
75 16cells  75 tesseracts  [5,3,3], order 14400 
300 16cells  300 tesseracts  [5,3,3]^{+}, order 7200 
600 16cells  600 tesseracts  [5,3,3], order 14400 
25 24cells  25 24cells  [5,3,3], order 14400 
Partially regular with convex 4polytopes:
Compound 1 Vertextransitive 
Compound 2 Celltransitive 
Symmetry 

2 16cells^{2}  2 tesseracts  [4,3,3], order 384 
100 24cell  100 24cell  [5,3,3]^{+}, order 7200 
200 24cell  200 24cell  [5,3,3], order 14400 
5 600cell  5 120cell  [5,3,3]^{+}, order 7200 
10 600cell  10 120cell  [5,3,3], order 14400 
Dual positions:
Compound  Symmetry 

2 5cell {{3,3,3}} 
[[3,3,3]], order 240 
2 24cell^{3} {{3,4,3}} 
[[3,4,3]], order 2304 
Compounds with regular star 4polytopes
Selfduals:
Compound  Symmetry 

5 {5,5/2,5}  [5,3,3]^{+}, order 7200 
10 {5,5/2,5}  [5,3,3], order 14400 
5 {5/2,5,5/2}  [5,3,3]^{+}, order 7200 
10 {5/2,5,5/2}  [5,3,3], order 14400 
Dual pairs:
Compound 1  Compound 2  Symmetry 

5 {3,5,5/2}  5 {5/2,5,3}  [5,3,3]^{+}, order 7200 
10 {3,5,5/2}  10 {5/2,5,3}  [5,3,3], order 14400 
5 {5,5/2,3}  5 {3,5/2,5}  [5,3,3]^{+}, order 7200 
10 {5,5/2,3}  10 {3,5/2,5}  [5,3,3], order 14400 
5 {5/2,3,5}  5 {5,3,5/2}  [5,3,3]^{+}, order 7200 
10 {5/2,3,5}  10 {5,3,5/2}  [5,3,3], order 14400 
Partially regular:
Compound 1 Vertextransitive 
Compound 2 Celltransitive 
Symmetry 

5 {3,3,5/2}  5 {5/2,3,3}  [5,3,3]^{+}, order 7200 
10 {3,3,5/2}  10 {5/2,3,3}  [5,3,3], order 14400 
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.
Footnotes
 ^ Richard Klitzing, Uniform compound, stellated icositetrachoron
 ^ Richard Klitzing, Uniform compound, demidistesseract
 ^ Richard Klitzing, Uniform compound, Dual positioned 24cells
External links
 MathWorld: Polyhedron Compound
 Stella: Polyhedron Navigator  Software used to create the images on this page. 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
 Compound of Small Stellated Dodecahedron and Great Dodecahedron {5/2,5}+{5,5/2}
 Compound polyhedra – from Virtual Reality
 Richard Klitzing, Compound polytopes, [2]
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. (c. 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 0486614808
 Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0520030567. p. 87 Five regular compounds
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