Spherical polyhedron

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The most familiar spherical polyhedron is the football, thought of as a spherical truncated icosahedron. Comparison of truncated icosahedron and soccer ball.png
The most 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. BeachBall.jpg
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. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.

Contents

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

The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period (the New Stone Age).

During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) wrote the first serious study of spherical polyhedra.

Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.

In the middle of the 20th Century, 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
configuration
pqq.2p.2pp.q.p.qp.2q.2qqpq.4.p.44.2q.2p3.3.q.3.p
Tetrahedral
symmetry
(3 3 2)
Uniform tiling 332-t0-1-.png
33
Uniform tiling 332-t01-1-.png
3.6.6
Uniform tiling 332-t1-1-.png
3.3.3.3
Uniform tiling 332-t12.png
3.6.6
Uniform tiling 332-t2.png
33
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 332-t012.png
4.6.6
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical triakis tetrahedron.png
V3.6.6
Spherical dual octahedron.png
V3.3.3.3
Spherical triakis tetrahedron.png
V3.6.6
Spherical rhombic dodecahedron.png
V3.4.3.4
Spherical tetrakis hexahedron.png
V4.6.6
Uniform tiling 532-t0.png
V3.3.3.3.3
Octahedral
symmetry
(4 3 2)
Uniform tiling 432-t0.png
43
Uniform tiling 432-t01.png
3.8.8
Uniform tiling 432-t1.png
3.4.3.4
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 432-t2.png
34
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 432-t012.png
4.6.8
Spherical snub cube.png
3.3.3.3.4
Spherical triakis octahedron.png
V3.8.8
Spherical rhombic dodecahedron.png
V3.4.3.4
Spherical tetrakis hexahedron.png
V4.6.6
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Spherical disdyakis dodecahedron.png
V4.6.8
Spherical pentagonal icositetrahedron.png
V3.3.3.3.4
Icosahedral
symmetry
(5 3 2)
Uniform tiling 532-t0.png
53
Uniform tiling 532-t01.png
3.10.10
Uniform tiling 532-t1.png
3.5.3.5
Uniform tiling 532-t12.png
5.6.6
Uniform tiling 532-t2.png
35
Uniform tiling 532-t02.png
3.4.5.4
Uniform tiling 532-t012.png
4.6.10
Spherical snub dodecahedron.png
3.3.3.3.5
Spherical triakis icosahedron.png
V3.10.10
Spherical rhombic triacontahedron.png
V3.5.3.5
Spherical pentakis dodecahedron.png
V5.6.6
Spherical deltoidal hexecontahedron.png
V3.4.5.4
Spherical disdyakis triacontahedron.png
V4.6.10
Spherical pentagonal hexecontahedron.png
V3.3.3.3.5
Dihedral
example p=6
(2 2 6)
Hexagonal dihedron.png
62
Dodecagonal dihedron.png
2.12.12
Hexagonal dihedron.png
2.6.2.6
Spherical hexagonal prism.svg
6.4.4
Hexagonal Hosohedron.svg
26
Spherical truncated trigonal prism.png
2.4.6.4
Spherical truncated hexagonal prism.png
4.4.12
Spherical hexagonal antiprism.svg
3.3.3.6
Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted). Sphere5tesselation.gif
Tiling of the sphere by spherical triangles (icosahedron with some of its spherical triangles distorted).
n234567810...
n-Prism
(2 2 p)
Tetragonal dihedron.png Spherical triangular prism.svg Spherical square prism2.png Spherical pentagonal prism.svg Spherical hexagonal prism2.png Spherical heptagonal prism.svg Spherical octagonal prism2.png Spherical decagonal prism2.png ...
n-Bipyramid
(2 2 p)
Spherical digonal bipyramid2.svg Spherical trigonal bipyramid.svg Spherical square bipyramid2.svg Spherical pentagonal bipyramid.svg Spherical hexagonal bipyramid2.png Spherical heptagonal bipyramid.svg Spherical octagonal bipyramid2.png Spherical decagonal bipyramid2.png ...
n-Antiprism Spherical digonal antiprism.svg Spherical trigonal antiprism.svg Spherical square antiprism.svg Spherical pentagonal antiprism.svg Spherical hexagonal antiprism.svg Spherical heptagonal antiprism.svg Spherical octagonal antiprism.svg ...
n-Trapezohedron Spherical digonal antiprism.svg Spherical trigonal trapezohedron.svg Spherical tetragonal trapezohedron.svg Spherical pentagonal trapezohedron.svg Spherical hexagonal trapezohedron.svg Spherical heptagonal trapezohedron.svg Spherical octagonal trapezohedron.svg Spherical decagonal trapezohedron.svg ...

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.

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn
SpaceSphericalEuclidean
Tiling name(Monogonal)
Henagonal hosohedron
Digonal hosohedron (Triangular)
Trigonal hosohedron
(Tetragonal)
Square hosohedron
Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedronOctagonal hosohedronEnneagonal hosohedronDecagonal hosohedronHendecagonal hosohedronDodecagonal hosohedron... Apeirogonal hosohedron
Tiling image Spherical henagonal hosohedron.svg Spherical digonal hosohedron.svg Spherical trigonal hosohedron.svg Spherical square hosohedron.svg Spherical pentagonal hosohedron.svg Spherical hexagonal hosohedron.svg Spherical heptagonal hosohedron.svg Spherical octagonal hosohedron.svg Spherical enneagonal hosohedron.svg Spherical decagonal hosohedron.svg Spherical hendecagonal hosohedron.svg Spherical dodecagonal hosohedron.svg ... Apeirogonal hosohedron.svg
Schläfli symbol {2,1}{2,2}{2,3}{2,4}{2,5}{2,6}{2,7}{2,8}{2,9}{2,10}{2,11}{2,12}...{2,∞}
Coxeter diagram CDel node 1.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 7.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 8.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 9.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 10.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 11.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 12.pngCDel node.png...CDel node 1.pngCDel 2x.pngCDel node.pngCDel infin.pngCDel node.png
Faces and edges123456789101112...
Vertices2...2
Vertex config. 22.223242526272829210211212...2
Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
SpaceSphericalEuclidean
Tiling name(Hengonal)
Monogonal dihedron
Digonal dihedron (Triangular)
Trigonal dihedron
(Tetragonal)
Square dihedron
Pentagonal dihedron Hexagonal dihedron ... Apeirogonal dihedron
Tiling image Monogonal dihedron.svg Digonal dihedron.svg Trigonal dihedron.svg Tetragonal dihedron.svg Pentagonal dihedron.svg Hexagonal dihedron.svg ... Apeirogonal tiling.svg
Schläfli symbol {1,2}{2,2}{3,2}{4,2}{5,2}{6,2}...{∞,2}
Coxeter diagram CDel node 1.pngCDel 1x.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png...CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png
Faces2 {1} 2 {2} 2 {3} 2 {4} 2 {5} 2 {6} ...2 {∞}
Edges and vertices123456...
Vertex config. 1.12.23.34.45.56.6...∞.∞

Relation to tilings of the projective plane

Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra [1] (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: [2]

See also

Related Research Articles

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<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

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In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.

<span class="mw-page-title-main">4-polytope</span> Four-dimensional geometric object with flat sides

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<span class="mw-page-title-main">Stellation</span> Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

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<span class="mw-page-title-main">Hosohedron</span> Spherical polyhedron composed of lunes

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<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

<span class="mw-page-title-main">Truncation (geometry)</span> Operation that cuts polytope vertices, creating a new facet in place of each vertex

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<span class="mw-page-title-main">Digon</span> Polygon with 2 sides and 2 vertices

In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space.

<span class="mw-page-title-main">Dihedron</span> Polyhedron with 2 faces

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In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

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<span class="mw-page-title-main">Regular 4-polytope</span> Four-dimensional analogues of the regular polyhedra in three dimensions

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.

In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.

References

  1. McMullen, Peter; Schulte, Egon (2002). "6C. Projective Regular Polytopes". Abstract Regular Polytopes. Cambridge University Press. pp.  162–5. ISBN   0-521-81496-0.
  2. 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.

Further reading