Spherical polyhedron


In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
Some [|"improper"] polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, is a hosohedron and is the dual dihedron.

History

The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period.
During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī wrote the first serious study of spherical polyhedra.
Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.
In the middle of the 20th Century, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes.

Examples

All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings. Given by their Schläfli symbol or vertex figure a.b.c....:
Schläfli symboltrtrrtrsr
Vertex figurepqq.2p.2pp.q.p.qp. 2q.2qqpq.4.p. 44.2q.2p3.3.q.3.p
Tetrahedral

33
3.6.6
3.3.3.3
3.6.6
33
3.4.3.4
4.6.6
3.3.3.3.3
Tetrahedral

33
V3.6.6
V3.3.3.3
V3.6.6
33
V3.4.3.4
V4.6.6
V3.3.3.3.3
Octahedral

43
3.8.8
3.4.3.4
4.6.6
34
3.4.4.4
4.6.8
3.3.3.3.4
Octahedral

43
V3.8.8
V3.4.3.4
V4.6.6
34
V3.4.4.4
V4.6.8
V3.3.3.3.4
Icosahedral

53
3.10.10
3.5.3.5
5.6.6
35
3.4.5.4
4.6.10
3.3.3.3.5
Icosahedral

53
V3.10.10
V3.5.3.5
V5.6.6
35
V3.4.5.4
V4.6.10
V3.3.3.3.5
Dihedral
example p=6

62
2.12.12
2.6.2.6
6.4.4
26
4.6.4
4.4.12
3.3.3.6

Class234567810
Prism
Bipyramid
Antiprism
Trapezohedron

Improper cases

Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as.
Image
Schläflih=...
Coxeter
Faces2 2 2 2 2 2
Edges and
vertices		123456

Relation to tilings of the projective plane

Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra: