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Spherical polyhedron

Partition of a sphere's surface into polygons

Partition of a sphere's surface into polygons

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.{{cite journal

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:

Schläflisymbol{p,q}t{p,q}r{p,q}t{q,p}{q,p}rr{p,q}tr{p,q}sr{p,q}Vertexconfig.pqq.2p.2pp.q.p.qp.2q.2qqpq.4.p.44.2q.2p3.3.q.3.pTetrahedralsymmetry(3 3 2)Octahedralsymmetry(4 3 2)Icosahedralsymmetry(5 3 2)Dihedralexample
(p=6)(2 2 6)
[[Image:Uniform tiling 332-t0-1-.svg64px]]33[[File:Uniform tiling 332-t01-1-.svg64px]]3.6.6[[Image:Uniform tiling 332-t1-1-.svg64px]]3.3.3.3[[Image:Uniform tiling 332-t12.svg64px]]3.6.6[[Image:Uniform tiling 332-t2.svg64px]]33[[Image:Uniform tiling 332-t02.svg64px]]3.4.3.4[[Image:Uniform tiling 332-t012.svg64px]]4.6.6[[Image:Spherical snub tetrahedron.svg64px]]3.3.3.3.3
[[Image:Spherical_triakis_tetrahedron.svg64px]]V3.6.6[[File:spherical_dual_octahedron.svg64px]]V3.3.3.3[[Image:Spherical_triakis_tetrahedron.svg64px]]V3.6.6[[File:Spherical rhombic dodecahedron.svg64px]]V3.4.3.4[[Image:Spherical tetrakis hexahedron.svg64px]]V4.6.6[[File:Uniform tiling 532-t0.svg64px]]V3.3.3.3.3
[[Image:Uniform tiling 432-t0.svg64px]]43[[Image:Uniform tiling 432-t01.svg64px]]3.8.8[[Image:Uniform tiling 432-t1.svg64px]]3.4.3.4[[Image:Uniform tiling 432-t12.svg64px]]4.6.6[[Image:Uniform tiling 432-t2.svg64px]]34[[Image:Uniform tiling 432-t02.svg64px]]3.4.4.4[[Image:Uniform tiling 432-t012.svg64px]]4.6.8[[Image:Spherical snub cube.svg64px]]3.3.3.3.4
[[Image:Spherical_triakis_octahedron.svg64px]]V3.8.8[[File:spherical_rhombic_dodecahedron.svg64px]]V3.4.3.4[[Image:Spherical tetrakis hexahedron.svg64px]]V4.6.6[[File:Spherical_deltoidal icositetrahedron.svg64px]]V3.4.4.4[[File:Spherical_disdyakis_dodecahedron1.svg64px]]V4.6.8[[File:Spherical_pentagonal_icositetrahedron.svg64px]]V3.3.3.3.4
[[Image:Uniform tiling 532-t0.svg64px]]53[[Image:Uniform tiling 532-t01.svg64px]]3.10.10[[Image:Uniform tiling 532-t1.png64px]]3.5.3.5[[Image:Uniform tiling 532-t12.png64px]]5.6.6[[Image:Uniform tiling 532-t2.svg64px]]35[[Image:Uniform tiling 532-t02.png64px]]3.4.5.4[[Image:Uniform tiling 532-t012.png64px]]4.6.10[[File:Spherical snub dodecahedron.png64px]]3.3.3.3.5
[[Image:Spherical_triakis_icosahedron.svg64px]]V3.10.10[[File:spherical_rhombic_triacontahedron.svg64px]]V3.5.3.5[[Image:Spherical_pentakis_dodecahedron.png64px]]V5.6.6[[File:Spherical_deltoidal hexecontahedron.svg64px]]V3.4.5.4[[File:Spherical_disdyakis_triacontahedron.svg64px]]V4.6.10[[File:Spherical_pentagonal_hexecontahedron.svg64px]]V3.3.3.3.5
[[Image:hexagonal dihedron.png64px]]62[[Image:dodecagonal dihedron.png64px]]2.12.12[[Image:hexagonal dihedron.png64px]]2.6.2.6[[File:Spherical hexagonal prism.svg64px]]6.4.4[[File:Hexagonal Hosohedron.svg64px]]26[[File:Spherical truncated trigonal prism.svg64px]]2.4.6.4[[File:Spherical truncated hexagonal prism.svg64px]]4.4.12[[File:Spherical hexagonal antiprism.svg64px]]3.3.3.6
Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted).
n234567...n-Prism(2 2 p)n-Bipyramid(2 2 p)n-Antiprismn-Trapezohedron
[[Image:Tetragonal dihedron.svg70px]][[Image:Spherical triangular prism.svg70px]][[Image:Spherical square prism2.svg70px]][[Image:Spherical pentagonal prism.svg70px]][[Image:Spherical hexagonal prism2.svg70px]][[Image:Spherical heptagonal prism.svg70px]]...
[[Image:Spherical digonal bipyramid2.svg70px]][[File:Spherical_trigonal_bipyramid.svg70px]][[Image:Spherical square bipyramid2.svg70px]][[File:Spherical pentagonal bipyramid.svg70px]][[Image:Spherical hexagonal bipyramid2.svg70px]][[File:Spherical_heptagonal_bipyramid.svg70px]]...
[[Image:Spherical digonal antiprism.svg70px]][[Image:Spherical trigonal antiprism.svg70px]][[Image:Spherical square antiprism.svg70px]][[Image:Spherical pentagonal antiprism.svg70px]][[Image:Spherical hexagonal antiprism.svg70px]][[Image:Spherical heptagonal antiprism.svg70px]]...
[[File:Spherical digonal antiprism.svg70px]][[File:Spherical trigonal trapezohedron.svg70px]][[File:Spherical tetragonal trapezohedron.svg70px]][[File:Spherical pentagonal trapezohedron.svg70px]][[File:Spherical hexagonal trapezohedron.svg70px]][[File:Spherical heptagonal trapezohedron.svg70px]]...

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 (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

References

References

  1. Popko, Edward S.. (2012). "Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere". CRC Press.
  2. (1954). "Uniform polyhedra". Phil. Trans..
  3. (2002). "Abstract Regular Polytopes". Cambridge University Press.
  4. Coxeter, H.S.M.. (1969). "Introduction to Geometry". Wiley.
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