Spherical polyhedron

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. 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: ==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 ==See also==