Spherical polyhedron
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.[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) |
64px 33 |
64px 3.6.6 |
64px 3.3.3.3 |
64px 3.6.6 |
64px 33 |
64px 3.4.3.4 |
64px 4.6.6 |
64px 3.3.3.3.3 |
| 64px V3.6.6 |
64px V3.3.3.3 |
64px V3.6.6 |
64px V3.4.3.4 |
64px V4.6.6 |
64px V3.3.3.3.3 | |||
| Octahedral symmetry (4 3 2) |
64px 43 |
64px 3.8.8 |
64px 3.4.3.4 |
64px 4.6.6 |
64px 34 |
64px 3.4.4.4 |
64px 4.6.8 |
64px 3.3.3.3.4 |
| 64px V3.8.8 |
64px V3.4.3.4 |
64px V4.6.6 |
64px V3.4.4.4 |
64px V4.6.8 |
64px V3.3.3.3.4 | |||
| Icosahedral symmetry (5 3 2) |
64px 53 |
64px 3.10.10 |
64px 3.5.3.5 |
64px 5.6.6 |
64px 35 |
64px 3.4.5.4 |
64px 4.6.10 |
64px 3.3.3.3.5 |
| 64px V3.10.10 |
64px V3.5.3.5 |
64px V5.6.6 |
64px V3.4.5.4 |
64px V4.6.10 |
64px V3.3.3.3.5 | |||
| Dihedral example (p=6) (2 2 6) |
64px 62 |
64px 2.12.12 |
64px 2.6.2.6 |
64px 6.4.4 |
64px 26 |
64px 2.4.6.4 |
64px 4.4.12 |
64px 3.3.3.6 |
| n | 2 | 3 | 4 | 5 | 6 | 7 | ... |
|---|---|---|---|---|---|---|---|
| n-Prism (2 2 p) |
70px | 70px | 70px | 70px | 70px | 70px | ... |
| n-Bipyramid (2 2 p) |
70px | 70px | 70px | 70px | 70px | 70px | ... |
| n-Antiprism | 70px | 70px | 70px | 70px | 70px | 70px | ... |
| n-Trapezohedron | 70px | 70px | 70px | 70px | 70px | 70px | ... |
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.
Template:Regular hosohedral tilings
Template:Regular dihedral tilings
Relation to tilings of the projective plane
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra[4] (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:[5]
- 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. https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PR19. ""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.
- ↑ 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. https://archive.org/details/introductiontoge00coxe.
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