Polytope 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 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
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.| Regular compound | Picture | Spherical | Up: Convex hull | Symmetry group | Subgroup restricting to one constituent | Dual |
| Two tetrahedra | Cube | *432 Oh | *332 Td | Self-dual | ||
| Five tetrahedra | Dodecahedron | 532 + I | 332 + T | Enantiomorph chiral twin | ||
| Ten tetrahedra 22 | Dodecahedron | *532 Ih | 332 T | Self-dual | ||
| Five cubes 2 | Dodecahedron | *532 Ih | 3*2 Th | Five octahedra | ||
| Five octahedra 2 | Icosidodecahedron | *532 Ih | 3*2 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 the compound of two tetrahedra 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 or "chiral-twin-dual", and the compound of 5 cubes is dual to the compound of 5 octahedra.
The five octahedra defining any given icosahedron form a regular polyhedral compound.
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 ce 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 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. For the convex solids, this is the convex hull.
| Dual compound | Picture | Hull | Core | Symmetry group |
| Two tetrahedra | Cube | Octahedron | *432 Oh | |
| Cube-octahedron | Rhombic dodecahedron | Cuboctahedron | *432 Oh | |
| Dodecahedron-icosahedron | Rhombic triacontahedron | Icosidodecahedron | *532 Ih | |
| Small stellated dodecahedron-great dodecahedron | Medial rhombic triacontahedron | Dodecadodecahedron | *532 Ih | |
| Great icosahedron-great stellated dodecahedron | Great rhombic triacontahedron | Great icosidodecahedron | *532 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
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
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
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 |
Compounds with duals
Dual positions:| Compound | Constituent | Symmetry |
| 2 5-cell | 5-cell | |
| 2 24-cell | 24-cell | |
| 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
In terms of group theory, if G is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra, 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 are analogous to the spherical stella octangula, 2.
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.