Polyhedral Compound
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Polyhedral 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 its convex hull. A compound is a facetting 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. There are five regular compounds of polyhedra:

Regular compound
(Coxeter symbol)
Picture Spherical Convex hull Common core Symmetry group Subgroup
restricting
to one
constituent
Dual-regular compound
Two tetrahedra
{4,3}[2{3,3}]{3,4}
Compound of two tetrahedra.png Spherical compound of two tetrahedra.png Cube

[1]

Octahedron *432
[4,3]
Oh
*332
[3,3]
Td
Two tetrahedra
Five tetrahedra
{5,3}[5{3,3}]{3,5}
Compound of five tetrahedra.png Spherical compound of five tetrahedra.png Dodecahedron

[1]

Icosahedron

[1]

532
[5,3]+
I
332
[3,3]+
T
Chiral twin
(Enantiomorph)
Ten tetrahedra
2{5,3}[10{3,3}]2{3,5}
Compound of ten tetrahedra.png Spherical compound of ten tetrahedra.png Dodecahedron

[1]

Icosahedron *532
[5,3]
Ih
332
[3,3]
T
Ten tetrahedra
Five cubes
2{5,3}[5{4,3}]
Compound of five cubes.png Spherical compound of five cubes.png Dodecahedron

[1]

Rhombic triacontahedron

[1]

*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

[1]

Icosahedron

[1]

*532
[5,3]
Ih
3*2
[3,3]
Th
Five cubes

Best known is the regular 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 define a regular octahedron, which shares the same face-planes as the compound. Thus the compound of two tetrahedra is a stellation of the octahedron, and in fact, the only finite stellation thereof.

The regular compound of five tetrahedra comes in two enantiomorphic versions, which together make up the regular compound of ten tetrahedra.[1] The regular compound of ten tetrahedra can also be constructed with five Stellae octangulae.[1]

Each of the regular tetrahedral compounds is self-dual or dual to its chiral twin; the regular compound of five cubes and the regular compound of five octahedra are dual to each other.

Hence, regular polyhedral compounds can also be regarded as dual-regular compounds.

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 {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.[2]

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 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.

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

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

Compound of 4 cubes.png Compound of 4 octahedra.png
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.

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

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[3]. McMullen added six in his paper New Regular Compounds of 4-Polytopes[4].

Self-duals:

Compound Constituent Symmetry
120 5-cells 5-cell [5,3,3], order 14400[3]
120 5-cells(var) 5-cell order 1200[4]
720 5-cells 5-cell [5,3,3], order 14400[3]
5 24-cells 24-cell [5,3,3], order 14400[3]

Dual pairs:

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

Uniform compounds and duals with convex 4-polytopes:

Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
2 16-cells[6] 2 tesseracts [4,3,3], order 384[3]
100 24-cells 100 24-cells [5,3,3]+, order 7200[3]
200 24-cells 200 24-cells [5,3,3], order 14400[3]
5 600-cells 5 120-cells [5,3,3]+, order 7200[3]
10 600-cells 10 120-cells [5,3,3], order 14400[3]
25 24-cells(var) 25 24-cells(var) order 600[4]

The superscript (var) in the tables above indicates that the labeled compounds are distinct from the other compounds with the same number of constituents.

Compounds with regular star 4-polytopes

Self-dual star compounds:

Compound Symmetry
5 {5,5/2,5} [5,3,3]+, order 7200[3]
10 {5,5/2,5} [5,3,3], order 14400[3]
5 {5/2,5,5/2} [5,3,3]+, order 7200[3]
10 {5/2,5,5/2} [5,3,3], order 14400[3]

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

Dual positions:

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 {∞,∞}
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

  1. ^ a b c d e f g h i j "Compound Polyhedra". www.georgehart.com. Retrieved .
  2. ^ Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. p. 48. ISBN 0-486-61480-8. OCLC 798003.
  3. ^ a b c d e f g h i j k l m n o p q r s Regular polytopes, Table VII, p. 305
  4. ^ a b c d McMullen, Peter (2018), New Regular Compounds of 4-Polytopes, New Trends in Intuitive Geometry, 27: 307-320
  5. ^ Klitzing, Richard. "Uniform compound stellated icositetrachoron".
  6. ^ Klitzing, Richard. "Uniform compound demidistesseract".

External links

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
  • McMullen, Peter (2018), "New Regular Compounds of 4-Polytopes", New Trends in Intuitive Geometry, 27: 307-320, doi:10.1007/978-3-662-57413-3_12.

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