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Polytope compound

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A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are 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-transitive, edge-transitive, and face-transitive. 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]
Oh 		*332
[3,3]
Td 		Self-dual
Compound of five tetrahedra, or chiro-icosahedron Dodecahedron Icosahedron 532
[5,3]+
I 		332
[3,3]+
T 		enantiomorph, or chiral twin
Compound of ten tetrahedra, compound of two chiro-icoshedra, or icosiicosahedron Dodecahedron Icosahedron *532
[5,3]
Ih 		332
[3,3]
T 		Self-dual
Compound of five cubes, or rhombihedron Dodecahedron Rhombic triacontahedron *532
[5,3]
Ih 		3*2
[3,3]
Th 		Compound of five octahedra
Compound of five octahedra, or small icosiicosahedron Icosidodecahedron Icosahedron *532
[5,3]
Ih 		3*2
[3,3]
Th 		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 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]
Oh
Compound of cube and octahedron Rhombic dodecahedron Cuboctahedron *432
[4,3]
Oh
Compound of dodecahedron and icosahedron Rhombic triacontahedron Icosidodecahedron *532
[5,3]
Ih
Compound of great icosahedron and great stellated dodecahedron Dodecahedron Icosidodecahedron *532
[5,3]
Ih
Compound of small stellated dodecahedron and great dodecahedron Icosahedron Dodecahedron *532
[5,3]
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. For this reason, the image shown above shows the small stellated dodecahedron in wireframe.

Uniform compounds

Main article: Uniform polyhedron compound

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

Other compounds

This compound of four cubes, though a polyhedral compound and despite its symmetries, is neither regular, nor dual-regular, nor a uniform compound.

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 polychoron compounds

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:

Self-duals:

Compound Symmetry
120 5-cell [5,3,3], order 14400
5 24-cell [5,3,3], order 14400

Dual pairs:

Compound 1 Compound 2 Symmetry
3 16-cells[1] 3 tesseracts [3,4,3], order 1152
15 16-cells 15 tesseracts [5,3,3], order 14400
75 16-cells 75 tesseracts [5,3,3], order 14400
300 16-cells 300 tesseracts [5,3,3]+, order 7200
600 16-cells 600 tesseracts [5,3,3], order 14400
25 24-cells 25 24-cells [5,3,3], order 14400

Partially regular with convex 4-polytopes:

Compound 1
Vertex-transitive 		Compound 2
Cell-transitive 		Symmetry
2 16-cells[2] 2 tesseracts [4,3,3], order 384
100 24-cell 100 24-cell [5,3,3]+, order 7200
200 24-cell 200 24-cell [5,3,3], order 14400
5 600-cell 5 120-cell [5,3,3]+, order 7200
10 600-cell 10 120-cell [5,3,3], order 14400

Dual positions:

Compound Symmetry
2 5-cell
{{3,3,3}} 		[[3,3,3]], order 240
2 24-cell[3]
{{3,4,3}} 		[[3,4,3]], order 2304

Compounds with regular star 4-polytopes

Self-duals:

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
Vertex-transitive 		Compound 2
Cell-transitive 		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

  1. ^ Klitzing, Richard. "Uniform compound stellated icositetrachoron".
  2. ^ Klitzing, Richard. "Uniform compound demidistesseract".
  3. ^ Klitzing, Richard. "Uniform compound Dual positioned 24-cells".

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{{citation}}: CS1 maint: location missing publisher (link).
  • 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 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
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