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.
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	Picture	Spherical	Convex hull	Common core	Symmetry group	Subgroup restricting to one constituent	Dual-regular compound
Two tetrahedra	100px	100px	Cube	Octahedron	*432 O_h	*332 T_d	Two tetrahedra
Five tetrahedra	100px	100px	Dodecahedron	Icosahedron	532 ⁺ I	332 ⁺ T	Chiral twin
Ten tetrahedra 22	100px	100px	Dodecahedron	Icosahedron	*532 I_h	332 T	Ten tetrahedra
Five cubes 2	100px	100px	Dodecahedron	Rhombic triacontahedron	*532 I_h	3*2 T_h	Five octahedra
Five octahedra 2	100px	100px	Icosidodecahedron	Icosahedron	*532 I_h	3*2 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. The regular compound of ten tetrahedra can also be seen as a compound of five stellae octangulae.
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, , denotes the components of the compound: d separate 's. The material before the square brackets denotes the vertex arrangement of the compound: c is a compound of d 's sharing the vertices of counted c times. The material after the square brackets denotes the facet arrangement of the compound: e is a compound of d 's sharing the faces of counted e times. These may be combined: thus c''e'' is a compound of d 's sharing the vertices of counted c times and the faces of counted e times. This notation can be generalised to compounds in any number of dimensions.

Dual compounds

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. For the convex solids, this is the convex hull.

Dual compound	Picture	Hull	Core	Symmetry group
Two tetrahedra	100px	Cube	Octahedron	*432 O_h
Cube and octahedron	100px	Rhombic dodecahedron	Cuboctahedron	*432 O_h
Dodecahedron and icosahedron	100px	Rhombic triacontahedron	Icosidodecahedron	*532 I_h
Small stellated dodecahedron and great dodecahedron		Medial rhombic triacontahedron	Dodecadodecahedron	*532 I_h
Great icosahedron and great stellated dodecahedron		Great rhombic triacontahedron	Great icosidodecahedron	*532 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 dual compound has the great dodecahedron completely interior to the small stellated dodecahedron.

Uniform compounds

In 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds made from uniform polyhedra with rotational symmetry. 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
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
Compound of three octahedra
Compound of four cubes

Two polyhedra that are compounds but have their elements rigidly locked into place are the small complex icosidodecahedron and the great complex icosidodecahedron. 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

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. McMullen added six in his paper New Regular Compounds of 4-Polytopes.
Self-duals:

Compound	Constituent	Symmetry
120 5-cells	5-cell	, order 14400
120 5-cells	5-cell	order 1200
720 5-cells	5-cell	, order 14400
5 24-cells	24-cell	, order 14400

Dual pairs:

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

Uniform compounds and duals with convex 4-polytopes:

Compound 1 Vertex-transitive	Compound 2 Cell-transitive	Symmetry
2 16-cells	2 tesseracts	, order 384
100 24-cells	100 24-cells	⁺, order 7200
200 24-cells	200 24-cells	, order 14400
5 600-cells	5 120-cells	⁺, order 7200
10 600-cells	10 120-cells	, order 14400
25 24-cells	25 24-cells	order 600

The superscript 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 Great 120-cell\|	⁺, order 7200
10 Great 120-cell\|	, order 14400
5 Grand stellated 120-cell\|	⁺, order 7200
10 Grand stellated 120-cell\|	, order 14400

Dual pairs of compound stars:

Compound 1	Compound 2	Symmetry
5	5	⁺, order 7200
10	10	, order 14400
5	5	⁺, order 7200
10	10	, order 14400
5	5	⁺, order 7200
10	10	, order 14400

Uniform compound stars and duals:

Compound 1 Vertex-transitive	Compound 2 Cell-transitive	Symmetry
5 Grand 600-cell\|	5 Great grand stellated 120-cell\|	⁺, order 7200
10 Grand 600-cell\|	10 Great grand stellated 120-cell\|	, order 14400