Polytope compound

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.^{[citation needed]}

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:

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: Icosadodecahedron)	*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. ^{[link removed]}

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)

[1]

[2]

[3]