- Hosohedron
An "n"-gonal

**hosohedron**is a tessellation of lunes on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron hasSchläfli symbol {2, "n"}.**Hosohedrons as regular polyhedrons**For a regular polyhedron whose Schläfli symbol is {"m", "n"}, the number of polygonal faces may be found by

:$N\_2=frac\{4n\}\{2m+2n-mn\}.$

The

platonic solid s known to antiquity are the only integer solutions for "m" ≥ 3 and "n" ≥ 3. The restriction "m" ≥ 3 enforces that the polygonal faces must have at least three sides.When considering polyhedrons as a

spherical tiling , this restriction may be relaxed, since digons can be represented as spherical lunes, having non-zero area. Allowing "m" = 2 admits a new infinite class of regular polyhedrons, which are the hosohedrons. On a spherical surface, the polyhedron {2, "n"} is represented as n abutting lunes, with interior angles of 2π/"n". All these lunes share two common vertices.**Derivative polyhedrons**The dual of the n-gonal hosohedron {2, "n"} is the "n"-gonal

dihedron , {"n", 2}. The polyhedron {2,2} is self-dual, and is both a hosohedron and a dihedron.A hosohedron may be modified in the same manner as the other polyhedrons to produce a truncated variation. The truncated "n"-gonal hosohedron is the n-gonal prism.

**Hosotopes**Multidimensional analogues in general are called

**hosotopes**, with Schläfli symbol "{2,...,2,q}". A hosotope has two vertices.The two-dimensional hosotope {2} is a

digon .**Etymology**The prefix “hoso-” was invented by H.S.M. Coxeter, and possibly derives from the English “hose”.

**ee also***

Polyhedron

*Polytope **References*** Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8

*

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Dihedron**— Set of regular n gonal dihedrons Example hexagonal dihedron on a sphere Type Regular polyhedron or spherical tiling Faces 2 n gons Edges n Vertices … Wikipedia**Schläfli symbol**— In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.The Schläfli symbol is named after the 19th century mathematician Ludwig Schläfli who made important contributions in… … Wikipedia**Pentahedron**— In geometry, a pentahedron (plural: pentahedra) is a polyhedron with five faces. Since there are no face transitive polyhedra with five sides and there are two distinct topological types, this term is rarely used.With regular polygon faces, the… … Wikipedia**Uniform polyhedron**— A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high… … Wikipedia**Digon**— A degenerate digon with two coinciding edges sharing the same vertices Edges and vertices 2 Schläfli symbol {2} … Wikipedia**Bipyramid**— For Dipyramid, see mountain and Dipyramid (Alaska). Set of bipyramids (Example hexagonal form) Faces 2n triangles Edges 3n Vertices n + 2 … Wikipedia**Cuboctahedron**— (Click here for rotating model) Type Archimedean solid Uniform polyhedron Elements F = 14, E = 24, V = 12 (χ = 2) Faces by sides 8{3}+6{4} … Wikipedia**Cube**— This article is about the geometric shape. For other uses, see Cube (disambiguation). Regular Hexahedron (Click here for rotating model) Type Platonic solid Elements F = 6, E = 12 V = 8 (χ = 2) … Wikipedia**Dodecahedron**— Regular Dodecahedron (Click here for rotating model) Type Platonic solid Elements F = 12, E = 30 V = 20 (χ = 2) Faces by sides 12{5} … Wikipedia**Octahedron**— For the album by The Mars Volta, see Octahedron (album). Regular Octahedron (Click here for rotating model) Type Platonic solid Elements F = 8, E = 12 V = 6 (χ = 2) Faces by sides … Wikipedia