Spherical polyhedron
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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 in Scotland, and appear to date from the neolithic period (the New Stone Age).
- 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).
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.
[edit] Examples
All the regular and semiregular polyhedra can be projected onto the sphere as tilings. Given by their Schläfli symbol {p, q} or vertex figure (a.b.c. ...):
| (p q r) | {p,q} | (q.2p.2p) | (p.q.p.q) | (p.2q.2q) | {q,p} | (q.4.p.4) | (2r.2q.2p) | (3.r.3.q.3.p) |
|---|---|---|---|---|---|---|---|---|
| Tetrahedral (3 3 2) |
 {3,3} |
 (3.6.6) |
 (3.3.3.3) |
 (3.6.6) |
 {3,3} |
 (3.4.3.4) |
 (4.6.6) |
 (3.2.3.3.3.3) |
| Octahedral (4 3 2) |
 {4,3} |
 (3.8.8) |
 (3.4.3.4) |
 (4.6.6) |
 {3,4} |
 (3.4.4.4) |
 (4.6.8) |
 (2.3.3.3.3.4) |
| Icosahedral (5 3 2) |
 {5,3} |
 (3.10.10) |
 (3.5.3.5) |
 (5.6.6) |
 {3,5} |
 (3.4.5.4) |
 (4.6.10) |
 (3.2.3.3.3.5) |
| Dihedral (6 2 2) example |
 {6,2} |
 (2.12.12) |
(2.6.2.6) |
 (6.4.4) |
 {2,6} |
 (2.4.6.4) |
 (4.4.12) |
 (3.2.3.2.3.6) |
[edit] See also
[edit] 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. [1]
