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.
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.
The work of Buckminster Fuller on geodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra. At roughly the same time, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).
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 |
| 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 hosohedra: figures as {2,n}, and dihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
| 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∞ |
| 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 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
References
- Sarhangi, Reza (September 2008). "Illustrating Abu al-Wafā' Būzjānī: Flat images, spherical constructions". Iranian Studies. 41 (4): 511–523. doi:10.1080/00210860802246184.
- Popko, Edward S. (2012). Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere. CRC Press. p. xix. ISBN 978-1-4665-0430-1.
Buckminster Fuller's invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development.
- Coxeter, H.S.M.; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra". Phil. Trans. 246 A (916): 401–50. JSTOR 91532.
- McMullen, Peter; Schulte, Egon (2002). "6C. Projective Regular Polytopes". Abstract Regular Polytopes. Cambridge University Press. pp. 162–5. ISBN 0-521-81496-0.
- Coxeter, H.S.M. (1969). "§21.3 Regular maps'". Introduction to Geometry (2nd ed.). Wiley. pp. 386–8. ISBN 978-0-471-50458-0. MR 0123930.
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 A familiar spherical polyhedron is the football thought of as a 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 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 HistoryDuring the 10th Century the Islamic scholar Abu al Wafa Buzjani Abu l Wafa studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architects The work of Buckminster Fuller on geodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra At roughly the same time Coxeter used them to enumerate all but one of the uniform polyhedra through the construction of kaleidoscopes Wythoff construction ExamplesAll regular polyhedra semiregular polyhedra and their duals can be projected onto the sphere as tilings Schlafli 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 pTetrahedral 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 3V3 6 6 V3 3 3 3 V3 6 6 V3 4 3 4 V4 6 6 V3 3 3 3 3Octahedral 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 4V3 8 8 V3 4 3 4 V4 6 6 V3 4 4 4 V4 6 8 V3 3 3 3 4Icosahedral 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 5V3 10 10 V3 5 3 5 V5 6 6 V3 4 5 4 V4 6 10 V3 3 3 3 5Dihedral 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 6Tiling 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 casesSpherical 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 Family of regular hosohedra n22 symmetry mutations of regular hosohedral tilings nn Space Spherical EuclideanTiling name Henagonal hosohedron Digonal hosohedron Trigonal hosohedron Square hosohedron Pentagonal hosohedron Apeirogonal hosohedronTiling image Schlafli 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 2Vertex config 2 2 2 23 24 25 2 Family of regular dihedra n22 symmetry mutations of regular dihedral tilings nn Space Spherical EuclideanTiling name Monogonal dihedron Digonal dihedron Trigonal dihedron Square dihedron Pentagonal dihedron Apeirogonal dihedronTiling image Schlafli 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 planeSpherical 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 gt 1 Hemi hosohedron 2 2p 2 p gt 1See alsoWikimedia Commons has media related to Spherical polyhedra Spherical geometry Spherical trigonometry Polyhedron Projective polyhedron Toroidal polyhedron Conway polyhedron notationReferencesSarhangi Reza September 2008 Illustrating Abu al Wafa Buzjani Flat images spherical constructions Iranian Studies 41 4 511 523 doi 10 1080 00210860802246184 Popko Edward S 2012 Divided Spheres Geodesics and the Orderly Subdivision of the Sphere CRC Press p xix ISBN 978 1 4665 0430 1 Buckminster Fuller s invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development Coxeter H S M Longuet Higgins M S Miller J C P 1954 Uniform polyhedra Phil Trans 246 A 916 401 50 JSTOR 91532 McMullen Peter Schulte Egon 2002 6C Projective Regular Polytopes Abstract Regular Polytopes Cambridge University Press pp 162 5 ISBN 0 521 81496 0 Coxeter H S M 1969 21 3 Regular maps Introduction to Geometry 2nd ed Wiley pp 386 8 ISBN 978 0 471 50458 0 MR 0123930
