Polytope compound

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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[edit]

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}
Compound of two tetrahedra.png Spherical compound of two tetrahedra.png Cube
(Octahedron)
*432
[4,3]
Oh
*332
[3,3]
Td
Self-dual
Five tetrahedra
{5,3}[5{3,3}]{3,5}
Compound of five tetrahedra.png Spherical compound of five tetrahedra.png Dodecahedron
(Icosahedron)
532
[5,3]+
I
332
[3,3]+
T
enantiomorph
chiral twin
Ten tetrahedra
2{5,3}[10{3,3}]2{3,5}
Compound of ten tetrahedra.png Spherical compound of ten tetrahedra.png Dodecahedron
(Icosahedron)
*532
[5,3]
Ih
332
[3,3]
T
Self-dual
Five cubes
2{5,3}[5{4,3}]
Compound of five cubes.png Spherical compound of five cubes.png Dodecahedron
(Rhombic triacontahedron)
*532
[5,3]
Ih
3*2
[3,3]
Th
Five octahedra
Five octahedra
[5{3,4}]2{3,5}
Compound of five octahedra.png Spherical compound of five octahedra.png 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.

Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, [d{p,q}], denotes the components of the compound: d separate {p,q}'s. The material before the square brackets denotes the vertex arrangement of the compound: c{m,n}[d{p,q}] is a compound of d {p,q}'s sharing the vertices of an {m,n} counted c times. The material after the square brackets denotes the facet arrangement of the compound: [d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the faces of {s,t} counted e times. These may be combined: thus c{m,n}[d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the vertices of {m,n} counted c times and the faces of {s,t} counted e times. This notation can be generalised to compounds in any number of dimensions.[1]

Dual compounds[edit]

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.

The core is the rectification of both solids. The hull is the dual of this rectification, and its faces have the intersecting edges of the two solids as diagonals. For the convex solids this is the convex hull.

Components Picture Hull Core Symmetry
Two tetrahedra
(Compound of two tetrahedra, stellated octahedron)
Dual compound 4 max.png Cube Octahedron *432
[4,3]
Oh
Cube, Octahedron
(Compound of cube and octahedron)
Dual compound 8 max.png Rhombic dodecahedron Cuboctahedron *432
[4,3]
Oh
Dodecahedron, Icosahedron
(Compound of dodecahedron and icosahedron)
Dual compound 20 max.png Rhombic triacontahedron Icosidodecahedron *532
[5,3]
Ih
Small stellated dodecahedron, Great dodecahedron
(Compound of sD and gD)
Skeleton pair Gr12 and dual, size m (crop), thick.png Medial rhombic triacontahedron
(Convex: Icosahedron)
Dodecadodecahedron
(Convex: Dodecahedron)
*532
[5,3]
Ih
Great icosahedron, Great stellated dodecahedron
(Compound of gI and gsD)
Skeleton pair Gr20 and dual, size s, thick.png Great rhombic triacontahedron
(Convex: Dodecahedron)
Great icosidodecahedron
(Convex: Icosahedron)
*532
[5,3]
Ih

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.

Uniform compounds[edit]

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]

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)
UC01-6 tetrahedra.png UC02-12 tetrahedra.png UC03-6 tetrahedra.png UC04-2 tetrahedra.png UC05-5 tetrahedra.png UC06-10 tetrahedra.png
UC07-6 cubes.png UC08-3 cubes.png UC09-5 cubes.png UC10-4 octahedra.png UC11-8 octahedra.png UC12-4 octahedra.png
UC13-20 octahedra.png UC14-20 octahedra.png UC15-10 octahedra.png UC16-10 octahedra.png UC17-5 octahedra.png UC18-5 tetrahemihexahedron.png
UC19-20 tetrahemihexahedron.png
UC20-2k n-m-gonal prisms.png UC21-k n-m-gonal prisms.png UC22-2k n-m-gonal antiprisms.png UC23-k n-m-gonal antiprisms.png UC24-2k n-m-gonal antiprisms.png UC25-k n-m-gonal antiprisms.png
UC26-12 pentagonal antiprisms.png UC27-6 pentagonal antiprisms.png UC28-12 pentagrammic crossed antiprisms.png UC29-6 pentagrammic crossed antiprisms.png UC30-4 triangular prisms.png UC31-8 triangular prisms.png
UC32-10 triangular prisms.png UC33-20 triangular prisms.png UC34-6 pentagonal prisms.png UC35-12 pentagonal prisms.png UC36-6 pentagrammic prisms.png UC37-12 pentagrammic prisms.png
UC38-4 hexagonal prisms.png UC39-10 hexagonal prisms.png UC40-6 decagonal prisms.png UC41-6 decagrammic prisms.png UC42-3 square antiprisms.png UC43-6 square antiprisms.png
UC44-6 pentagrammic antiprisms.png UC45-12 pentagrammic antiprisms.png
  • 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,
UC46-2 icosahedra.png UC47-5 icosahedra.png UC48-2 great dodecahedra.png UC49-5 great dodecahedra.png UC50-2 small stellated dodecahedra.png UC51-5 small stellated dodecahedra.png
UC52-2 great icosahedra.png UC53-5 great icosahedra.png UC54-2 truncated tetrahedra.png UC55-5 truncated tetrahedra.png UC56-10 truncated tetrahedra.png UC57-5 truncated cubes.png
UC58-5 quasitruncated hexahedra.png UC59-5 cuboctahedra.png UC60-5 cubohemioctahedra.png UC61-5 octahemioctahedra.png UC62-5 rhombicuboctahedra.png UC63-5 small rhombihexahedra.png
UC64-5 small cubicuboctahedra.png UC65-5 great cubicuboctahedra.png UC66-5 great rhombihexahedra.png UC67-5 great rhombicuboctahedra.png
UC68-2 snub cubes.png UC69-2 snub dodecahedra.png UC70-2 great snub icosidodecahedra.png UC71-2 great inverted snub icosidodecahedra.png UC72-2 great retrosnub icosidodecahedra.png UC73-2 snub dodecadodecahedra.png
UC74-2 inverted snub dodecadodecahedra.png UC75-2 snub icosidodecadodecahedra.png

Other compounds[edit]

Compound of 4 cubes.png Compound of 4 octahedra.png
These compounds, of four cubes, and (dual) four octahedra, are neither regular compounds, nor dual compounds, nor uniform 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.

4-polytope compounds[edit]

Orthogonal projections
Regular compound 75 tesseracts.png Regular compound 75 16-cells.png
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:[2]

Self-duals:

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

Dual pairs:

Compound 1 Compound 2 Symmetry
3 16-cells[3] 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

Uniform compounds and duals with convex 4-polytopes:

Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
2 16-cells[4] 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

Compounds with regular star 4-polytopes[edit]

Self-dual star compounds:

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 of compound stars:

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

Uniform compound stars and duals:

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

Compounds with duals[edit]

Dual positions:

Compound Constituent Symmetry
2 5-cell 5-cell [[3,3,3]], order 240
2 24-cell 24-cell [[3,4,3]], order 2304
1 tesseract, 1 16-cell tesseract, 16-cell
1 120-cell, 1 600-cell 120-cell, 600-cell
2 great 120-cell great 120-cell
2 grand stellated 120-cell grand stellated 120-cell
1 icosahedral 120-cell, 1 small stellated 120-cell icosahedral 120-cell, small stellated 120-cell
1 grand 120-cell, 1 great stellated 120-cell grand 120-cell, great stellated 120-cell
1 great grand 120-cell, 1 great icosahedral 120-cell great grand 120-cell, great icosahedral 120-cell
1 great grand stellated 120-cell, 1 grand 600-cell great grand stellated 120-cell, grand 600-cell

Only the first two of these dual compounds are also regular.

Group theory[edit]

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[edit]

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 {∞,∞}
Kah 4 4.png Compound 2 hexagonal tilings.png Compound 2 triangular tilings.png Infinite-order apeirogonal tiling and dual.png
3 {6,3} 3 {3,6} 3 {∞,∞}
Compound 3 hexagonal tilings.png Compound 3 triangular tilings.png Iii symmetry 000.png

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, which shares its edges with the deltoidal trihexagonal tiling. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.

Footnotes[edit]

  1. ^ Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. p. 48. ISBN 0-486-61480-8. OCLC 798003.
  2. ^ Regular polytopes, Table VII, p. 305
  3. ^ Klitzing, Richard. "Uniform compound stellated icositetrachoron".
  4. ^ Klitzing, Richard. "Uniform compound demidistesseract".

External links[edit]

References[edit]

  • 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