Hosohedron

Set of regular n-gonal hosohedrons
Example hexagonal hosohedron on a sphere
Type	Regular polyhedron or spherical tiling
Faces	n digons
Edges	n
Vertices	2
Schläfli symbol	{2,n}
Vertex configuration	2n
Coxeter–Dynkin diagram
Wythoff symbol	n | 2 2
Symmetry group	Dnh, [2,n], (*22n)
Dual polyhedron	dihedron

This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

In geometry, 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.

A regular trigonal hosohedron, represented as a tessellation of 3 spherical lunes on a sphere.

A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.

Relationship with Steinmetz Solid

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.

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. A regular hosotope with Schläfli symbol {2,p,...,q} has two vertices, each with a vertex figure {p,...,q}.

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
• Weisstein, Eric W., "Hosohedron" from MathWorld.