Hosohedron

Short description: Spherical polyhedron composed of lunes

Set of regular n-gonal hosohedra
Example regular hexagonal hosohedron on a sphere
Type	regular polyhedron or spherical tiling
Faces	n digons
Edges	n
Vertices	2
χ	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	regular n-gonal dihedron

This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.

In spherical 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/nradians (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 :

[math]\displaystyle{ N_2=\frac{4n}{2m+2n-mn}. }[/math]

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 makes

[math]\displaystyle{ N_2=\frac{4n}{2\times2+2n-2n}=n, }[/math]

and 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 spherical 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.

Kaleidoscopic symmetry

The [math]\displaystyle{ 2n }[/math] digonal spherical lune faces of a [math]\displaystyle{ 2n }[/math]-hosohedron, [math]\displaystyle{ \{2,2n\} }[/math], represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry [math]\displaystyle{ C_{nv} }[/math], [math]\displaystyle{ [n] }[/math], [math]\displaystyle{ (*nn) }[/math], order [math]\displaystyle{ 2n }[/math]. The reflection domains can be shown by alternately colored lunes as mirror images.

Bisecting each lune into two spherical triangles creates an [math]\displaystyle{ n }[/math]-gonal bipyramid, which represents the dihedral symmetry [math]\displaystyle{ D_{nh} }[/math], order [math]\displaystyle{ 4n }[/math].

Different representations of the kaleidoscopic symmetry of certain small hosohedra

Symmetry (order [math]\displaystyle{ 2n }[/math])	Schönflies notation	[math]\displaystyle{ C_{nv} }[/math]	[math]\displaystyle{ C_{1v} }[/math]	[math]\displaystyle{ C_{2v} }[/math]	[math]\displaystyle{ C_{3v} }[/math]	[math]\displaystyle{ C_{4v} }[/math]	[math]\displaystyle{ C_{5v} }[/math]	[math]\displaystyle{ C_{6v} }[/math]
	Orbifold notation	[math]\displaystyle{ (*nn) }[/math]	[math]\displaystyle{ (*11) }[/math]	[math]\displaystyle{ (*22) }[/math]	[math]\displaystyle{ (*33) }[/math]	[math]\displaystyle{ (*44) }[/math]	[math]\displaystyle{ (*55) }[/math]	[math]\displaystyle{ (*66) }[/math]
	Coxeter diagram
		[math]\displaystyle{ [n] }[/math]	[math]\displaystyle{ [\,\,] }[/math]	[math]\displaystyle{ [2] }[/math]	[math]\displaystyle{ [3] }[/math]	[math]\displaystyle{ [4] }[/math]	[math]\displaystyle{ [5] }[/math]	[math]\displaystyle{ [6] }[/math]
[math]\displaystyle{ 2n }[/math]-gonal hosohedron	Schläfli symbol	[math]\displaystyle{ \{2,2n\} }[/math]	[math]\displaystyle{ \{2,2\} }[/math]	[math]\displaystyle{ \{2,4\} }[/math]	[math]\displaystyle{ \{2,6\} }[/math]	[math]\displaystyle{ \{2,8\} }[/math]	[math]\displaystyle{ \{2,10\} }[/math]	[math]\displaystyle{ \{2,12\} }[/math]
	Alternately colored 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” appears to derive from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”. It was introduced by Vito Caravelli in the eighteenth century.

See also

• Polyhedron
• Polytope

References

• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0, https://archive.org/details/abstractregularp0000mcmu 
• Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc., ISBN:0-486-61480-8

External links

• Weisstein, Eric W.. "Hosohedron". http://mathworld.wolfram.com/Hosohedron.html.