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, {2, 6},Script error: No such module "Check for unknown parameters". is a hosohedron, and {6, 2} Script error: No such module "Check for unknown parameters". is its dual dihedron.
History
During the 10th Century, the Islamic scholar Abū 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 of Buckminster Fuller on geodesic 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 the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).[3]
Examples
All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
Improper cases
Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as {2,n}, and dihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
Template:Regular hosohedral tilings
Template:Regular dihedral tilings
Relation to tilings of the projective plane
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra[4] (tessellations of the real projective plane) – 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:[5]
- Hemi-cube, {4,3}/2
- Hemi-octahedron, {3,4}/2
- Hemi-dodecahedron, {5,3}/2
- Hemi-icosahedron, {3,5}/2
- Hemi-dihedron, {2p,2}/2, p≥1
- Hemi-hosohedron, {2,2p}/2, p≥1
See also
- Spherical geometry
- Spherical trigonometry
- Polyhedron
- Projective polyhedron
- Toroidal polyhedron
- Conway polyhedron notation
References
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