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

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 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”.

ee also

* Polyhedron
* Polytope

References

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