Spherical polyhedron

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The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron.
This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

In mathematics, a spherical polyhedron 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 (outside the USA, a football), thought of as a spherical truncated icosahedron.

Some polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, {2, 6} is a hosohedron and {6, 2} is the dual dihedron.

History[edit]

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 (the New Stone Age).

During the European "Dark Age", the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) 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 (Wythoff construction).

Examples[edit]

All the regular and semiregular polyhedra can be projected onto the sphere as tilings. Given by their Schläfli symbol {p, q} or vertex figure (a.b.c. ...):

(p q r) {p,q} (q.2p.2p) (p.q.p.q) (p. 2q.2q) {q,p} (q.4.p. 4) (2r.2q.2p) (3.r.3.q.3.p)
Tetrahedral
(3 3 2)
Uniform tiling 332-t0-1-.png
{3,3}
Uniform tiling 332-t01-1-.png
(3.6.6)
Uniform tiling 332-t1-1-.png
(3.3.3.3)
Uniform tiling 332-t12.png
(3.6.6)
Uniform tiling 332-t2.png
{3,3}
Uniform tiling 332-t02.png
(3.4.3.4)
Uniform tiling 332-t012.png
(4.6.6)
Spherical snub tetrahedron.png
(3.2.3.3.3.3)
Octahedral
(4 3 2)
Uniform tiling 432-t0.png
{4,3}
Uniform tiling 432-t01.png
(3.8.8)
Uniform tiling 432-t1.png
(3.4.3.4)
Uniform tiling 432-t12.png
(4.6.6)
Uniform tiling 432-t2.png
{3,4}
Uniform tiling 432-t02.png
(3.4.4.4)
Uniform tiling 432-t012.png
(4.6.8)
Spherical snub cube.png
(3.2.3.3.3.4)
Icosahedral
(5 3 2)
Uniform tiling 532-t0.png
{5,3}
Uniform tiling 532-t01.png
(3.10.10)
Uniform tiling 532-t1.png
(3.5.3.5)
Uniform tiling 532-t12.png
(5.6.6)
Uniform tiling 532-t2.png
{3,5}
Uniform tiling 532-t02.png
(3.4.5.4)
Uniform tiling 532-t012.png
(4.6.10)
Spherical snub dodecahedron.png
(3.2.3.3.3.5)
Dihedral
(6 2 2)
example
Hexagonal dihedron.png
{6,2}
Dodecagonal dihedron.png
(2.12.12)
Hexagonal dihedron.png
(2.6.2.6)
Spherical hexagonal prism.png
(6.4.4)
Hexagonal Hosohedron.svg
{2,6}
Spherical truncated trigonal prism.png
(2.4.6.4)
Spherical truncated hexagonal prism.png
(4.4.12)
Spherical hexagonal antiprism.png
(3.2.3.2.3.6)

Relation to tilings of the projective plane[edit]

Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra[1] (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. For example, the 2-fold cover of the (projective) hemi-cube is the (spherical) cube.

See also[edit]

References[edit]

  1. ^ McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0