- Spherical polyhedron
In

mathematics , the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tiling or**spherical polyhedron**. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.Spherical polyhedra have a long and respectable history:

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

*Two hundred years ago, at the start of the19th Century , Poinsot used spherical polyhedra to discover the four regular star polyhedra.

*In the middle of the20th Century , Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction ).The most familiar spherical polyhedron is the football (soccer ball), 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.**Examples**All the regular and semiregular polyhedra can be projected onto the sphere as tilings. Given by their

Schläfli symbol {p, q} orvertex figure (a.b.c. ...):**ee also***

Spherical geometry

*Spherical trigonometry

*Polyhedra **References*** L. Poinsot, Memoire sur les polygones et polyèdres. "J. de l'École Polytechnique"

**9**, (1810), pp. 16-48.

*H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, "Phil. Trans."**246 A**, (1954), pp. 401-50. [*http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4*]

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