Ingeometry, aspherical polyhedron orspherical tiling is atiling of thesphere in which the surface is divided or partitioned bygreat arcs into bounded regions calledspherical polygons. Apolyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron.
The most familiar spherical polyhedron is thesoccer ball, thought of as a sphericaltruncated icosahedron. The next most popular spherical polyhedron is thebeach ball, thought of as ahosohedron.
Some"improper" polyhedra, such ashosohedra and theirduals,dihedra, exist as spherical polyhedra, but their flat-faced analogs aredegenerate. The example hexagonal beach ball,{2, 6}, is a hosohedron, and{6, 2} is its dual dihedron.
During the 10th Century, the Islamic scholarAbū al-Wafā' Būzjānī (Abu'l Wafa) studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architects.[1]
The work ofBuckminster Fuller ongeodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra.[2] At roughly the same time,Coxeter used them to enumerate all but one of theuniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).[3]
Allregular polyhedra,semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
| Schläfli symbol | {p,q} | t{p,q} | r{p,q} | t{q,p} | {q,p} | rr{p,q} | tr{p,q} | sr{p,q} |
|---|---|---|---|---|---|---|---|---|
| Vertex config. | pq | q.2p.2p | p.q.p.q | p.2q.2q | qp | q.4.p.4 | 4.2q.2p | 3.3.q.3.p |
| Tetrahedral symmetry (3 3 2) |  33 |  3.6.6 |  3.3.3.3 |  3.6.6 |  33 |  3.4.3.4 |  4.6.6 |  3.3.3.3.3 |
 V3.6.6 |  V3.3.3.3 | V3.6.6 |  V3.4.3.4 |  V4.6.6 |  V3.3.3.3.3 | |||
| Octahedral symmetry (4 3 2) |  43 |  3.8.8 |  3.4.3.4 |  4.6.6 |  34 |  3.4.4.4 |  4.6.8 |  3.3.3.3.4 |
 V3.8.8 | V3.4.3.4 | V4.6.6 |  V3.4.4.4 |  V4.6.8 |  V3.3.3.3.4 | |||
| Icosahedral symmetry (5 3 2) | 53 |  3.10.10 |  3.5.3.5 |  5.6.6 |  35 |  3.4.5.4 |  4.6.10 |  3.3.3.3.5 |
 V3.10.10 |  V3.5.3.5 |  V5.6.6 |  V3.4.5.4 |  V4.6.10 |  V3.3.3.3.5 | |||
| Dihedral example (p=6) (2 2 6) |  62 |  2.12.12 | 2.6.2.6 |  6.4.4 |  26 |  2.4.6.4 |  4.4.12 |  3.3.3.6 |
| n | 2 | 3 | 4 | 5 | 6 | 7 | ... |
|---|---|---|---|---|---|---|---|
| n-Prism (2 2 p) |  |  |  |  |  |  | ... |
| n-Bipyramid (2 2 p) |  |  |  |  |  |  | ... |
| n-Antiprism |  |  |  |  | |  | ... |
| n-Trapezohedron | |  |  |  |  |  | ... |
Spherical tilings allow cases that polyhedra do not, namelyhosohedra: figures as {2,n}, anddihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
| Space | Spherical | Euclidean | |||||
|---|---|---|---|---|---|---|---|
| Tiling name | Henagonal hosohedron | Digonal hosohedron | Trigonal hosohedron | Square hosohedron | Pentagonal hosohedron | ... | Apeirogonal hosohedron |
| Tiling image |  |  |  |  |  | ... |  |
| Schläfli symbol | {2,1} | {2,2} | {2,3} | {2,4} | {2,5} | ... | {2,∞} |
| Coxeter diagram |    | |  |  |  | ... |  |
| Faces and edges | 1 | 2 | 3 | 4 | 5 | ... | ∞ |
| Vertices | 2 | 2 | 2 | 2 | 2 | ... | 2 |
| Vertex config. | 2 | 2.2 | 23 | 24 | 25 | ... | 2∞ |
| Space | Spherical | Euclidean | |||||
|---|---|---|---|---|---|---|---|
| Tiling name | Monogonal dihedron | Digonal dihedron | Trigonal dihedron | Square dihedron | Pentagonal dihedron | ... | Apeirogonal dihedron |
| Tiling image |  |  |  |  |  | ... |  |
| Schläfli symbol | {1,2} | {2,2} | {3,2} | {4,2} | {5,2} | ... | {∞,2} |
| Coxeter diagram |  | | | | | ... | |
| Faces | 2{1} | 2{2} | 2{3} | 2{4} | 2{5} | ... | 2{∞} |
| Edges and vertices | 1 | 2 | 3 | 4 | 5 | ... | ∞ |
| Vertex config. | 1.1 | 2.2 | 3.3 | 4.4 | 5.5 | ... | ∞.∞ |
Spherical polyhedra having at least oneinversive symmetry are related toprojective polyhedra[4] (tessellations of thereal projective plane) – just as the sphere has a 2-to-1covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric underreflection through the origin.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of thecentrally symmetricPlatonic solids, as well as two infinite classes of evendihedra andhosohedra:[5]
Buckminster Fuller's invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development.
