This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols for all m such that m < n/2 (strictly speaking =) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k as being more correct, where usually k = m.
A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k, as 2, rather than the commonly used .
Coxeter's extended notation for compounds is of the form c[''d''{''p'',''q'',...}]e, indicating that d distinct 's together cover the vertices of c times and the facets of e times. If no regular exists, the first part of the notation is removed, leaving [''d''{''p'',''q'',...}]e; the opposite holds if no regular exists. The dual of c[''d''{''p'',''q'',...}]e is e[''d''{''q'',''p'',...}]c. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to [''k''{''n''/''m''}]: for example, the hexagram may be written thus as [2{3}].
Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:
| Compound skew squares | Compound skew hexagons | Compound skew decagons | ||
| Two # | Three # | Two # | Two # | |
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.
| Symmetry | [4,3], Oh | [5,3]+, I | [5,3], Ih | |||
|---|---|---|---|---|---|---|
| Duality | Self-dual | Dual pairs | ||||
| Image | ||||||
| Spherical | ||||||
| Polyhedra | 2 | 5 | 10 | 5 | 5 | |
| Coxeter | [2[[tetrahedron|{3,3}]]] | [5[[tetrahedron|{3,3}]]] | 2[10[[tetrahedron|{3,3}]]]2 | 2[5[[cube|{4,3}]]] | [5[[octahedron|{3,4}]]]2 | |
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 's. The material before the square brackets denotes the vertex arrangement of the compound: c[''d''{''p'',''q''}] is a compound of d 's sharing the vertices of an counted c times. The material after the square brackets denotes the facet arrangement of the compound: [''d''{''p'',''q''}]e is a compound of d 's sharing the faces of counted e times. These may be combined: thus c[''d''{''p'',''q''}]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.
If improper regular polyhedra (dihedra and hosohedra) are allowed, then two more compounds are possible: 2[3{4,2}] and its dual [3{2,4}]2.[1]
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 yet been proven.[1]
| Self-dual | |||
|---|---|---|---|
| [(b<sup>2</sup>+c<sup>2</sup>){4,4}] | b ≥ c ≥ 0, b > 0 | ||
| 2[2(b<sup>2</sup>+c<sup>2</sup>){4,4}]2 | bc(b − c) > 0 | ||
| Dual pairs | |||
| [(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}]2 | 2[(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}] | b ≡ c mod 3 | |
| 2[2(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}]4 | 4[2(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}]2 | b ≡ c mod 3, bc(b − c) > 0 | |
| [2(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}]2 | 2[2(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}] | b ≡ c mod 3 | |
| 2[4(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}]4 | 4[4(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}]2 | b ≡ c mod 3, bc(b − c) > 0 | |
| [(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}] | [(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}] | b ≢ c mod 3 | |
| 2[2(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}]2 | 2[2(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}]2 | b ≢ c mod 3, bc(b − c) > 0 | |
| [2(b<sup>2</sup>+bc+c<sup>2</sup>){3,6}] | [2(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}] | b ≢ c mod 3 | |
| 2[4(b<sup>2</sup>+bc+c<sup>2</sup>)2{3,6}] | [4(b<sup>2</sup>+bc+c<sup>2</sup>){6,3}]2 | b ≢ c mod 3, bc(b − c) > 0 | |
A distinction must be made when an integer can be expressed in the forms b2+c2 or b2+bc+c2 in two different ways, e.g. 145 = 122 + 12 = 92 + 82, or 91 = 92 + 9 ⋅ 1 + 12 = 62 + 6 ⋅ 5 + 52. In such cases, Coxeter notates the sum explicitly, e.g. [(144+1){4,4}] as opposed to [(81+64){4,4}].[1]
The following compounds of compact or paracompact hyperbolic tessellations were known to Coxeter in 1964, though a proof of completeness was not then known:[1]
| [2{q,q}] | ||
| [6{8,8}] | ||
| [6{10,10}] | ||
| [12{10,10}] | ||
| [9{7,7}] | ||
| 2[18{7,7}]2 | ||
| Dual pairs | ||
|---|---|---|
| [3{q,2q}]2 | 2[3{2q,q}] | |
| [6{4,10}]2 | 2[6{10,4}] | |
| [8{3,14}]2 | 2[8{14,3}] | |
| [24{7,14}]2 | 2[24{14,7}] | |
| [12{9,18}]2 | 2[12{18,9}] | |
| [2{q,2q}] | [2{2q,q}] | |
| 2[9{4,7}] | [9{7,4}]2 | |
| [4{3,18}] | [4{18,3}] | |
The Euclidean and hyperbolic compound families [2{q,q}] appear because h =, i.e. taking alternate vertices of a results in a . They are thus the Euclidean and hyperbolic analogues of the spherical stella octangula, which is the q=3 case.[1]
It is also the case that h =, yielding the compound [2{q,2q}] and its dual [2{2q,q}]. Now if we take the dual of the, we obtain a third whose vertices are at the centres of alternate faces of the other two ; this gives the compound [3{q,2q}]2 and its dual 2[3{2q,q}]. These compounds are hyperbolic if q > 3 and Euclidean if q = 3. These compounds show an analogy to the spherical compounds [2{3,3,4}], [2{4,3,3}], [3{3,3,4}]2, and 2[3{4,3,3}].[1]
If one sets q = 8 in [2{q,q}], and q = 4 in [3{q,2q}]2, then one obtains the special cases [2{8,8}] and [3{4,8}]2. The latter's 's can be replaced by pairs of 's according to the former, giving the self-dual compound [6{8,8}].[1]