Spherical polyhedron

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 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. 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 United States, Canada, and Australia, a football), 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 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

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

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)	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		V3.4.4.4	V4.6.6	V3.3.3.3.3
Octahedral (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 (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	4.6.4	4.4.12	3.3.3.6

Class	2	3	4	5	6	7	8	10
Prism (2 2 p)
Bipyramid (2 2 p)
Antiprism
Trapezohedron

Improper cases

Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as {2,n}, and dihedra, regular figures as {n,2}.

Family of regular hosohedra

Image
Schläfli	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	{2,6}	{2,7}	{2,8}...
Coxeter
Faces and edges	1	2	3	4	5	6	7	8
Vertices	2

Regular dihedra: (spherical tilings)

Image
Schläfli	h{2,2}={1,2}	{2,2}	{3,2}	{4,2}	{5,2}	{6,2}...
Coxeter
Faces	2 {1}	2 {2}	2 {3}	2 {4}	2 {5}	2 {6}
Edges and vertices	1	2	3	4	5	6

Relation to tilings of the projective plane

Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra (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:

• 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

Further reading

• L. Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, (1810), pp. 16–48.
• H.S.M. Coxeter, M.S. Longuet-Higgins, J. C. P. Miller, Uniform polyhedra, Phil. Trans. 246 A, (1954), pp. 401–50. [1]
• H.S.M. Coxeter, Regular polytopes, 3rd Ed., Dover (1973).