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 has Schläfli symbol {2, n}.

Hosohedrons as regular polyhedrons

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

The platonic solids 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”.

See also

• Polyhedron
• Polytope

References

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