Spherical polyhedron


In geometry, 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. A polyhedron 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 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 hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron.

History

During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architects.
The work of Buckminster Fuller on geodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra. At roughly the same time, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes.

Examples

All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
Schläfli
symbol		trtrrtrsr
Vertex
config.		pqq.2p.2pp.q.p.qp.2q.2qqpq.4.p.44.2q.2p3.3.q.3.p
Tetrahedral
symmetry
64px
33
3.6.6		64px
3.3.3.3		64px
3.6.6		64px
33		64px
3.4.3.4		64px
4.6.6		64px
3.3.3.3.3
Tetrahedral
symmetry
64px
33		64px
V3.6.6
V3.3.3.3		64px
V3.6.6		64px
33
V3.4.3.4		64px
V4.6.6
V3.3.3.3.3
Octahedral
symmetry
64px
43		64px
3.8.8		64px
3.4.3.4		64px
4.6.6		64px
34		64px
3.4.4.4		64px
4.6.8		64px
3.3.3.3.4
Octahedral
symmetry
64px
43		64px
V3.8.8
V3.4.3.4		64px
V4.6.6		64px
34
V3.4.4.4
V4.6.8
V3.3.3.3.4
Icosahedral
symmetry
64px
53		64px
3.10.10		64px
3.5.3.5		64px
5.6.6		64px
35		64px
3.4.5.4		64px
4.6.10
3.3.3.3.5
Icosahedral
symmetry
64px
53		64px
V3.10.10
V3.5.3.5		64px
V5.6.6		64px
35
V3.4.5.4
V4.6.10
V3.3.3.3.5
Dihedral
example

64px
62		64px
2.12.12		64px
2.6.2.6
6.4.4
26
2.4.6.4
4.4.12
3.3.3.6

n234567...
n-Prism
70px70px70px70px70px70px...
n-Bipyramid
70px70px70px...
n-Antiprism70px70px70px70px70px70px...
n-Trapezohedron...

Improper cases

Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as, and dihedra: figures as. Generally, regular hosohedra and regular dihedra are used.

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: