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 and Australia, a football), thought of as a spherical truncated icosahedron.

Some "improper" 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, semiregular polyhedra and their duals can be projected onto the sphere as tilings. Given by their Schläfli symbol {p, q} or vertex figure a.b.c. ...:

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 figure 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
(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
33
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 332-t012.png
4.6.6
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical triakis tetrahedron.png
V3.6.6
Spherical dual octahedron.png
V3.3.3.3
Spherical triakis tetrahedron.png
V3.6.6
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Spherical tetrakis hexahedron.png
V4.6.6
Uniform tiling 532-t2.png
V3.3.3.3.3
Octahedral
(4 3 2)
Uniform tiling 432-t0.png
43
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
34
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 432-t012.png
4.6.8
Spherical snub cube.png
3.3.3.3.4
Spherical triakis octahedron.png
V3.8.8
Spherical rhombic dodecahedron.png
V3.4.3.4
Spherical tetrakis hexahedron.png
V4.6.6
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Spherical disdyakis dodecahedron.png
V4.6.8
Spherical pentagonal icositetrahedron.png
V3.3.3.3.4
Icosahedral
(5 3 2)
Uniform tiling 532-t0.png
53
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
35
Uniform tiling 532-t02.png
3.4.5.4
Uniform tiling 532-t012.png
4.6.10
Spherical snub dodecahedron.png
3.3.3.3.5
Spherical triakis icosahedron.png
V3.10.10
Spherical rhombic triacontahedron.png
V3.5.3.5
Spherical pentakis dodecahedron.png
V5.6.6
Spherical deltoidal hexecontahedron.png
V3.4.5.4
Spherical disdyakis triacontahedron.png
V4.6.10
Spherical pentagonal hexecontahedron.png
V3.3.3.3.5
Dihedral
example p=6
(2 2 6)
Hexagonal dihedron.png
62
Dodecagonal dihedron.png
2.12.12
Hexagonal dihedron.png
2.6.2.6
Spherical hexagonal prism.png
6.4.4
Hexagonal Hosohedron.svg
26
Spherical truncated trigonal prism.png
4.6.4
Spherical truncated hexagonal prism.png
4.4.12
Spherical hexagonal antiprism.png
3.3.3.6
Class 2 3 4 5 6 7 8 10
Prism
(2 2 p)
Spherical triangular prism.png Spherical square prism2.png Spherical pentagonal prism.png Spherical hexagonal prism2.png Spherical heptagonal prism.png Spherical octagonal prism2.png Spherical decagonal prism2.png
Bipyramid
(2 2 p)
Spherical digonal bipyramid2.png Spherical trigonal bipyramid.png Spherical square bipyramid2.png Spherical pentagonal bipyramid.png Spherical hexagonal bipyramid2.png Spherical heptagonal bipyramid.png Spherical octagonal bipyramid2.png Spherical decagonal bipyramid2.png
Antiprism Spherical digonal antiprism.png Spherical trigonal antiprism.png Spherical square antiprism.png Spherical pentagonal antiprism.png Spherical hexagonal antiprism.png Spherical heptagonal antiprism.png Spherical octagonal antiprism.png
Trapezohedron Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png Spherical decagonal trapezohedron.png

Improper cases[edit]

Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as {2,n}, and dihedra, regular figures as {n,2}. Both families have limits with n=1 represent single vertex or single face figures.

Family of regular hosohedra
Image Spherical henagonal hosohedron.png Spherical digonal hosohedron.png Spherical trigonal hosohedron.png Spherical square hosohedron.png Spherical pentagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical heptagonal hosohedron.png Spherical octagonal hosohedron.png
Schläfli {2,1} {2,2} {2,3} {2,4} {2,5} {2,6} {2,7} {2,8}...
Coxeter CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node h0.png = CDel node 1.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 7.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 8.pngCDel node.png
Faces and
edges
1 2 3 4 5 6 7 8
Vertices 2
Regular dihedrons: (spherical tilings)
Image Hengonal dihedron.png Digonal dihedron.png Trigonal dihedron.png Tetragonal dihedron.png Pentagonal dihedron.png Hexagonal dihedron.png
Schläfli {1,2} = h{2,2} {2,2} {3,2} {4,2} {5,2} {6,2}...
Coxeter CDel node h.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png
Faces 2 {1} 2 {2} 2 {3} 2 {4} 2 {5} 2 {6}
Edges and
vertices
1 2 3 4 5 6

The simplest (and trivial) regular tiling of the sphere is {1,1}, a monohedron, consists of a single vertex, no edges and a single face as the sphere outside the vertex. It is self-dual, i.e. the point and face center can be swapped creating itself as a central inversion.

Limiting cases
Self-dual Dual tilings Self-dual
Image Digonal dihedron.pngSpherical digonal hosohedron.png Hengonal dihedron.png Spherical henagonal hosohedron.png Spherical henagonal henahedron.png
Schläfli {2,2} {1,2} = h{2,2} {2,1} {1,1} = h{2,1}
Coxeter CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node h.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node h0.png = CDel node 1.pngCDel 2x.pngCDel node.png CDel node h.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node h0.png = CDel node h.pngCDel 2x.pngCDel node.png
Faces 2 2 1 1
Edges 2 1 1 0
Vertices 2 1 2 1

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.

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:[2]

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 
  2. ^ Coxeter, Introduction to geometry, 1969, Second edition, sec 21.3 Regular maps, p. 386-388

Further reading[edit]