In geometry, 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 its convex hull. A compound is a faceting of its 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 polyhedral compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its flags; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra:
{| class="wikitable"
!Regular compound<BR>(Coxeter symbol)
!Picture
!Spherical
!Convex hull
!Common core
!Symmetry group
!Subgroup<br>restricting<br>to one<br>constituent
!Dual-regular compound
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| Two tetrahedra<BR>{4,3}[2{3,3}]{3,4}
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||Cube
The regular compound of ten tetrahedra can also be seen as a compound of five stellae octangulae.
Dual compounds
A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five dual compounds of the regular polyhedra.
The core is the rectification of both solids. The hull is the dual of this rectification, and its rhombic faces have the intersecting edges of the two solids as diagonals (and have their four alternate vertices). For the convex solids, this is the convex hull.
{| class="wikitable"
! Dual compound
! Picture
! Hull
! Core
!Symmetry group
|-
| Two tetrahedra<br><small>(Compound of two tetrahedra, stellated octahedron)</small>
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|Cube
|Octahedron
|style="text-align:center"|*432<br>[4,3]<br>O<sub>h</sub>
|-
| Cube and octahedron<br><small>(Compound of cube and octahedron)</small>
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| Rhombic dodecahedron
| Cuboctahedron
|style="text-align:center"|*432<br>[4,3]<br>O<sub>h</sub>
|-
| Dodecahedron and icosahedron<br><small>(Compound of dodecahedron and icosahedron)</small>
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| Rhombic triacontahedron
| Icosidodecahedron
|style="text-align:center"|*532<br>[5,3]<br>I<sub>h</sub>
|-
| Small stellated dodecahedron and great dodecahedron<br><small>(Compound of sD and gD)</small>
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| Medial rhombic triacontahedron<br><small>(Convex: Icosahedron)</small>
| Dodecadodecahedron<br><small>(Convex: Dodecahedron)</small>
|style="text-align:center"|*532<br>[5,3]<br>I<sub>h</sub>
|-
| Great icosahedron and great stellated dodecahedron<br><small>(Compound of gI and gsD)</small>
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| Great rhombic triacontahedron<br><small>(Convex: Dodecahedron)</small>
| Great icosidodecahedron<br><small>(Convex: Icosadodecahedron)</small>
|style="text-align:center"|*532<br>[5,3]<br>I<sub>h</sub>
|}
The tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual is the regular stellated octahedron.
The octahedral and icosahedral dual compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively.
The small stellated dodecahedral (or great dodecahedral) dual compound has the great dodecahedron completely interior to the small stellated dodecahedron.
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. [https://archive.today/20070928154042/http://www.interocitors.com/polyhedra/UCs/UniformCompounds.html]
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)
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- 20-25: Prism symmetry embedded in prism symmetry,
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- 26-45: Prism symmetry embedded in octahedral or icosahedral symmetry,
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- 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
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- 68-75: enantiomorph pairs
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Other compounds
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|colspan=2|The compound of four cubes (left) is neither a regular compound, nor a dual compound, nor a uniform compound. Its dual, the compound of four octahedra (right), is a uniform compound.
|}
- 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 enantiomorph 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
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|+ Orthogonal projections
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!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 these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes.
Self-duals:
{| class=wikitable
!Compound
!Constituent
!Symmetry
|-
| 120 5-cells || 5-cell || [5,3,3], order 14400||3 tesseracts||[3,4,3], order 1152||2 tesseracts||[4,3,3], order 384