Spherical polyhedron

Source: Wikipedia, the free encyclopedia.
spherical truncated icosahedron
.
This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed.

In

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}, 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:

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
Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted).
n 2 3 4 5 6 7 ...
n-Prism
(2 2 p) 		...
n-Bipyramid
(2 2 p) 		...
n-Antiprism ...
n-Trapezohedron ...

Improper cases

Spherical tilings allow cases that polyhedra do not, namely

dihedra
: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
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∞
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
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 ... ∞.∞

Relation to tilings of the projective plane

Spherical polyhedra having at least one

reflection through the origin
.

The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the

hosohedra:[5]

See also

Wikimedia Commons has media related to Spherical polyhedra.

References

  1. doi:10.1080/00210860802246184
    .
  2. ISBN 978-1-4665-0430-1
    . Buckminster Fuller's invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development.
  3. JSTOR 91532
    .
  4. ISBN 0-521-81496-0
    .
  5. MR 0123930
    .
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