Polytope compound

Last updated December 09, 2024

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.

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:

Regular compound (Coxeter symbol)	Convex hull	Common core	Symmetry group	Subgroup restricting to one constituent	Dual-regular compound
Two tetrahedra {4,3}[2{3,3}]{3,4}	Cube ^[1]	Octahedron	*432 [4,3] O_h	*332 [3,3] T_d	Two tetrahedra
Five tetrahedra {5,3}[5{3,3}]{3,5}	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}	Dodecahedron ^[1]	Icosahedron	*532 [5,3] I_h	332 [3,3] T	Ten tetrahedra
Five cubes 2{5,3}[5{4,3}]	Dodecahedron ^[1]	Rhombic triacontahedron ^[1]	*532 [5,3] I_h	3*2 [3,3] T_h	Five octahedra
Five octahedra [5{3,4}]2{3,5}	Icosidodecahedron ^[1]	Icosahedron ^[1]	*532 [5,3] I_h	3*2 [3,3] T_h	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 seen as a compound of 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

Truncated tetrahedron (light) and triakis tetrahedron (dark)

Snub cube (light) and pentagonal icositetrahedron (dark)

Icosidodecahedron (light) and rhombic triacontahedron (dark)

Dual compounds of Archimedean and Catalan solids

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.

Dual compound	Hull	Core	Symmetry group
Two tetrahedra (Compound of two tetrahedra, stellated octahedron)	Cube	Octahedron	*432 [4,3] O_h
Cube and octahedron (Compound of cube and octahedron)	Rhombic dodecahedron	Cuboctahedron	*432 [4,3] O_h
Dodecahedron and icosahedron (Compound of dodecahedron and icosahedron)	Rhombic triacontahedron	Icosidodecahedron	*532 [5,3] I_h
Small stellated dodecahedron and great dodecahedron (Compound of sD and gD)	Medial rhombic triacontahedron (Convex: Icosahedron)	Dodecadodecahedron (Convex: Dodecahedron)	*532 [5,3] I_h
Great icosahedron and great stellated dodecahedron (Compound of gI and gsD)	Great rhombic triacontahedron (Convex: Dodecahedron)	Great icosidodecahedron (Convex: Icosahedron)	*532 [5,3] I_h

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.^[3]

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.

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)

20-25: Prism symmetry embedded in prism symmetry,

26-45: Prism symmetry embedded in octahedral or icosahedral symmetry,

46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,

68-75: enantiomorph pairs

Other compounds


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
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.^[4] McMullen added six in his paper New Regular Compounds of 4-Polytopes.^[5]

Self-duals:

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

Dual pairs:

Compound 1	Compound 2	Symmetry
3 16-cells ^[6]	3 tesseracts	[3,4,3], order 1152^[4]
15 16-cells	15 tesseracts	[5,3,3], order 14400^[4]
75 16-cells	75 tesseracts	[5,3,3], order 14400^[4]
75 16-cells ^(var)	75 tesseracts ^(var)	order 600^[5]
300 16-cells	300 tesseracts	[5,3,3]⁺, order 7200^[4]
600 16-cells	600 tesseracts	[5,3,3], order 14400^[4]
25 24-cells	25 24-cells	[5,3,3], order 14400^[4]

Uniform compounds and duals with convex 4-polytopes:

Compound 1 Vertex-transitive	Compound 2 Cell-transitive	Symmetry
2 16-cells ^[7]	2 tesseracts	[4,3,3], order 384^[4]
100 24-cells	100 24-cells	[5,3,3]⁺, order 7200^[4]
200 24-cells	200 24-cells	[5,3,3], order 14400^[4]
5 600-cells	5 120-cells	[5,3,3]⁺, order 7200^[4]
10 600-cells	10 120-cells	[5,3,3], order 14400^[4]
25 24-cells ^(var)	25 24-cells ^(var)	order 600^[5]

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^[4]
10 {5,5/2,5}	[5,3,3], order 14400^[4]
5 {5/2,5,5/2}	[5,3,3]⁺, order 7200^[4]
10 {5/2,5,5/2}	[5,3,3], order 14400^[4]

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:

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

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 {∞,∞}

	3 {6,3}	3 {3,6}	3 {∞,∞}

A known family of regular Euclidean compound honeycombs in any number of 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 2 3 4 5 6 7 8 9 10 "Compound Polyhedra". www.georgehart.com. Retrieved 2020-09-03.
↑ Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. p. 48. ISBN 0-486-61480-8. OCLC 798003.
↑ "Great Dodecahedron-Small Stellated Dodecahedron Compound".
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Regular polytopes, Table VII, p. 305
1 2 3 4 McMullen, Peter (2018), New Regular Compounds of 4-Polytopes, New Trends in Intuitive Geometry, 27: 307–320
↑ Klitzing, Richard. "Uniform compound stellated icositetrachoron".
↑ Klitzing, Richard. "Uniform compound demidistesseract".

External links

Related Research Articles

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

<span class="mw-page-title-main">Polyhedron</span> Three-dimensional shape with flat faces, straight edges, and sharp corners

In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the Bilinski dodecahedron. There are some stellations of the rhombic dodecahedron, one of which is the Escher's solid. The rhombic dodecahedron may also appear in the garnet crystal, the architectural philosophies, practical usages, and toys.

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.

In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.

<span class="mw-page-title-main">Faceting</span> Removing parts of a polytope without creating new vertices

In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

<span class="mw-page-title-main">Regular 4-polytope</span> Four-dimensional analogues of the regular polyhedra in three dimensions

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.

References

Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society , 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554, S2CID 123279687 .
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. (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, Bolyai Society Mathematical Studies, vol. 27, pp. 307–320, doi:10.1007/978-3-662-57413-3_12, ISBN 978-3-662-57412-6 .

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[:0-1] 1 2 3 4 5 6 7 8 9 10 "Compound Polyhedra". www.georgehart.com. Retrieved 2020-09-03.

[2] Coxeter, Harold Scott MacDonald (1973) [1948]. Regular Polytopes (Third ed.). Dover Publications. p. 48. ISBN 0-486-61480-8. OCLC 798003.

[3] "Great Dodecahedron-Small Stellated Dodecahedron Compound".

[cox-4] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Regular polytopes, Table VII, p. 305

[McMullen-5] 1 2 3 4 McMullen, Peter (2018), New Regular Compounds of 4-Polytopes, New Trends in Intuitive Geometry, 27: 307–320

[6] Klitzing, Richard. "Uniform compound stellated icositetrachoron".

[7] Klitzing, Richard. "Uniform compound demidistesseract".

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