Hosohedron

Set of regular n-gonal hosohedra
Example hexagonal hosohedron on a sphere
Type	Regular polyhedron or spherical tiling
Faces	n digons
Edges	n
Vertices	2
Euler char.	2
Vertex configuration	2n
Wythoff symbol	n | 2 2
Schläfli symbol	{2,n}
Coxeter diagram
Symmetry group	Dnh, [2,n], (*22n), order 4n
Rotation group	Dn, [2,n]+, (22n), order 2n
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 polar opposite vertices.

A regular n-gonal hosohedron has Schläfli symbol {2, n}, with each spherical lune having internal angle 2π/n radians (360/n degrees).

Hosohedra as regular polyhedra

Further information: List_of_regular_polytopes_and_compounds § Spherical_2

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :

$N_{2}={\frac {4n}{2m+2n-mn}}$

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 polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. 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, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.

A regular tetragonal hosohedron, {2,4}, represented as a tessellation of 4 spherical lunes on a sphere.

Family of regular (n-gonal) hosohedra (2 vertices)

n	1	2	3	4	5	6	7	8	9	10	11	12...
n-gonal hosohedron image
Schläfli symbol {2,n}	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	{2,6}	{2,7}	{2,8}	{2,9}	{2,10}	{2,11}	{2,12}
Coxeter diagram

Kaleidoscopic symmetry

The digonal (lune) faces of a 2n-hosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: C_nv, [n], (*nn), order 2n. The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry D_nh, order 4n.

Symmetry	C1v, [ ]	C2v, [2]	C3v, [3]	C4v, [4]	C5v, [5]	C6v, [6]
Hosohedron	{2,2}	{2,4}	{2,6}	{2,8}	{2,10}	{2,12}
Fundamental domains

Relationship with the Steinmetz solid

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

Derivative polyhedra

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 polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism.

Apeirogonal hosohedron

In the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

Hosotopes

Further information: List_of_regular_polytopes_and_compounds § Spherical_3

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 term “hosohedron” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”.

See also

• Polyhedron
• Polytope

References

• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0
• Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8

External links

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