Spherical polyhedron

{{short description|Partition of a sphere's surface into polygons}} [[File:Comparison of truncated icosahedron and soccer ball.png|280px|thumb|right|A familiar spherical polyhedron is the football, thought of as a spherical truncated icosahedron.]] [[Image:BeachBall.jpg|thumb|This beach ball 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.<ref>{{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}}</ref>

The work of Buckminster Fuller on geodesic domes in the mid 20th century triggered a boom in the study of spherical polyhedra.<ref>{{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."}}</ref> At roughly the same time, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).<ref>{{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}}</ref>

==Examples== All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings: {| class="wikitable" !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 |-align=center !rowspan=2|Tetrahedral symmetry (3 3 2) |rowspan=2|64px 33 |64px 3.6.6 |64px 3.3.3.3 |64px 3.6.6 |rowspan=2|64px 33 |64px 3.4.3.4 |64px 4.6.6 |64px 3.3.3.3.3 |-align=center |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 |-align=center !rowspan=2|Octahedral symmetry (4 3 2) |rowspan=2|64px 43 |64px 3.8.8 |64px 3.4.3.4 |64px 4.6.6 |rowspan=2|64px 34 |64px 3.4.4.4 |64px 4.6.8 |64px 3.3.3.3.4 |-align=center |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 |-align=center !rowspan=2|Icosahedral symmetry (5 3 2) |rowspan=2|64px 53 |64px 3.10.10 |64px 3.5.3.5 |64px 5.6.6 |rowspan=2|64px 35 |64px 3.4.5.4 |64px 4.6.10 |64px 3.3.3.3.5 |-align=center |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 |- align=center !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 |}

thumb|Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted).

{| 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== 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.

==Relation to tilings of the projective plane== Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra<ref>{{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] }}</ref> (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:<ref name="cox">{{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}}</ref> * 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}} *Spherical geometry *Spherical trigonometry *Polyhedron *Projective polyhedron *Toroidal polyhedron *Conway polyhedron notation

==References== {{reflist}}

Category:Polyhedra Category:Tessellation Category:Spheres