spherical polyhedron
{{short description|Partition of a sphere's surface into polygons}}
File:Comparison of truncated icosahedron and soccer ball.png, thought of as a spherical truncated icosahedron.]]
Image:BeachBall.jpg would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed.]]
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, {{math|{2, 6},}} is a hosohedron, and {{math|{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.{{cite journal
| last = Sarhangi | first = Reza
| date = September 2008
| doi = 10.1080/00210860802246184
| issue = 4
| journal = Iranian Studies
| pages = 511–523
| title = Illustrating Abu al-Wafā' Būzjānī: Flat images, spherical constructions
| volume = 41}}
The work of Buckminster Fuller on geodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra.{{cite book|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=978-1-4665-0430-1|page=xix|url=https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PR19|quote="Buckminster Fuller’s invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development."}} At roughly the same time, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).{{cite journal |author-link=Harold Scott MacDonald Coxeter |first1=H.S.M. |last1=Coxeter |author2-link=Michael S. Longuet-Higgins |first2=M.S. |last2=Longuet-Higgins |author3-link=J. C. P. Miller |first3=J.C.P. |last3=Miller |title=Uniform polyhedra |journal=Phil. Trans. |volume=246 A |issue= 916|pages=401–50 |year=1954 |jstor=91532}}
Examples
All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
| class="wikitable"
!{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 |
|---|
| align=center
!rowspan=2|Tetrahedral |
| align=center |
| align=center
!rowspan=2|Octahedral |
| align=center |
| align=center
!rowspan=2|Icosahedral |
| align=center |
| align=center
!Dihedral |
| class=wikitable
!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
Relation to tilings of the projective plane
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra{{cite book | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | chapter = 6C. Projective Regular Polytopes | title = Abstract Regular Polytopes | publisher = Cambridge University Press | isbn = 0-521-81496-0 |date=2002 | pages = [https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA162 162–5] }} (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:{{cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | title=Introduction to Geometry | url=https://archive.org/details/introductiontoge00coxe | url-access=limited | publisher=Wiley | edition=2nd | isbn=978-0-471-50458-0 | mr=123930 | year=1969 |chapter=§21.3 Regular maps' |pages=[https://archive.org/details/introductiontoge00coxe/page/n403 386]–8}}
- 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
{{Commonscat|Spherical polyhedra}}