Spherical polyhedron

Partition of a sphere's surface into polygons From Wikipedia, the free encyclopedia

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.

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A familiar spherical polyhedron is the football, thought of as a spherical truncated icosahedron.
Thumb
This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed.

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}, is a hosohedron, and {6, 2} 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:

More information Schläflisymbol, {p,q} ...
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)
Thumb
33
Thumb
3.6.6
Thumb
3.3.3.3
Thumb
3.6.6
Thumb
33
Thumb
3.4.3.4
Thumb
4.6.6
Thumb
3.3.3.3.3
Thumb
V3.6.6
Thumb
V3.3.3.3
Thumb
V3.6.6
Thumb
V3.4.3.4
Thumb
V4.6.6
Thumb
V3.3.3.3.3
Octahedral
symmetry
(4 3 2)
Thumb
43
Thumb
3.8.8
Thumb
3.4.3.4
Thumb
4.6.6
Thumb
34
Thumb
3.4.4.4
Thumb
4.6.8
Thumb
3.3.3.3.4
Thumb
V3.8.8
Thumb
V3.4.3.4
Thumb
V4.6.6
Thumb
V3.4.4.4
Thumb
V4.6.8
Thumb
V3.3.3.3.4
Icosahedral
symmetry
(5 3 2)
Thumb
53
Thumb
3.10.10
Thumb
3.5.3.5
Thumb
5.6.6
Thumb
35
Thumb
3.4.5.4
Thumb
4.6.10
Thumb
3.3.3.3.5
Thumb
V3.10.10
Thumb
V3.5.3.5
Thumb
V5.6.6
Thumb
V3.4.5.4
Thumb
V4.6.10
Thumb
V3.3.3.3.5
Dihedral
example
(p=6)
(2 2 6)
Thumb
62
Thumb
2.12.12
Thumb
2.6.2.6
Thumb
6.4.4
Thumb
26
Thumb
2.4.6.4
Thumb
4.4.12
Thumb
3.3.3.6
Close
Thumb
Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted).
More information n, ... ...
n 2 3 4 5 6 7 ...
n-Prism
(2 2 p)
Thumb Thumb Thumb Thumb Thumb Thumb ...
n-Bipyramid
(2 2 p)
Thumb Thumb Thumb Thumb Thumb Thumb ...
n-Antiprism Thumb Thumb Thumb Thumb Thumb Thumb ...
n-Trapezohedron Thumb Thumb Thumb Thumb Thumb Thumb ...
Close

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.

More information Space, Euclidean ...
Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
Space SphericalEuclidean
Tiling
name
Henagonal
hosohedron
Digonal
hosohedron
Trigonal
hosohedron
Square
hosohedron
Pentagonal
hosohedron
... Apeirogonal
hosohedron
Tiling
image
Thumb Thumb Thumb Thumb Thumb ... Thumb
Schläfli
symbol
{2,1}{2,2}{2,3}{2,4}{2,5}...{2,∞}
Coxeter
diagram
...
Faces and
edges
12345...
Vertices 22222...2
Vertex
config.
22.2232425...2
Close
More information Space, Euclidean ...
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
Space SphericalEuclidean
Tiling
name
Monogonal
dihedron
Digonal
dihedron
Trigonal
dihedron
Square
dihedron
Pentagonal
dihedron
... Apeirogonal
dihedron
Tiling
image
Thumb Thumb Thumb Thumb Thumb ... Thumb
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
12345...
Vertex
config.
1.12.23.34.45.5...∞.∞
Close

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]

See also

References

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