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Spherical polyhedron

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Title: Spherical polyhedron  
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Subject: Spherical trigonometry, Bipyramid, Gilbert tessellation, Tiling with rectangles, Order-5 square tiling
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Spherical polyhedron

The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron.
This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

In mathematics, a spherical polyhedron 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.

The most familiar spherical polyhedron is the soccer ball (outside the USA, a football), thought of as a spherical truncated icosahedron.

Some polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, {2, 6} is a hosohedron and {6, 2} is the 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 European "Dark Age", 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 the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings. Given by their Schläfli symbol {p, q} or vertex figure a.b.c. ...:
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 figure 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
(3 3 2)

{3,3}

3.6.6

3.3.3.3

3.6.6

33

3.4.3.4

4.6.6

3.3.3.3.3

V3.6.6

V3.3.3.3

V3.6.6

V3.4.4.4

V4.6.6

V3.3.3.3.3
Octahedral
(4 3 2)

43

3.8.8

3.4.3.4

4.6.6

34

3.4.4.4

4.6.8

3.3.3.3.4

V3.8.8

V3.4.3.4

V4.6.6

V3.4.4.4

V4.6.8

V3.3.3.3.4
Icosahedral
(5 3 2)

53

3.10.10

3.5.3.5

5.6.6

35

3.4.5.4

4.6.10

3.3.3.3.5

V3.10.10

V3.5.3.5

V5.6.6

V3.4.5.4

V4.6.10

V3.3.3.3.5
Dihedral
example
(6 2 2)

62

2.12.12

2.6.2.6

6.4.4

26

4.6.4

4.4.12

3.3.3.6
Class 2 3 4 5 6 7 8 10
Prism
Bipyramid
Antiprism
Trapezohedron

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. For example, the 2-fold cover of the (projective) hemi-cube is the (spherical) cube.

See also

References

Further reading

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