Hosohedron

Set of regular n-gonal hosohedra
File:Hexagonal Hosohedron.svg Example regular 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	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel n.pngFile:CDel node.png
Symmetry group	Dnh [2,n] (*22n) order 4n
Rotation group	Dn [2,n]+ (22n) order 2n
Dual polyhedron	regular n-gonal dihedron

File:BeachBall.jpg

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 :

$N_2=\frac{4n}{2m+2n-mn}.$

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 makes

$N_2=\frac{4n}{2\times 2+2n-2n}=n,$

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.

File:Trigonal hosohedron.png A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.

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

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn

Space	Spherical						Euclidean
Tiling name	Henagonal hosohedron	Digonal hosohedron	Trigonal hosohedron	Square hosohedron	Pentagonal hosohedron	...	Apeirogonal hosohedron
Tiling image	File:Spherical henagonal hosohedron.svg	File:Spherical digonal hosohedron.svg	File:Spherical trigonal hosohedron.svg	File:Spherical square hosohedron.svg	File:Spherical pentagonal hosohedron.svg	...	File:Apeirogonal hosohedron.svg
Schläfli symbol	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	...	{2,∞}
Coxeter diagram	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel 2x.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	...	File:CDel node 1.pngFile:CDel 2x.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
Faces and edges	1	2	3	4	5	...	∞
Vertices	2	2	2	2	2	...	2
Vertex config.	2	2.2	23	24	25	...	2∞

Kaleidoscopic symmetry

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

Bisecting each lune into two spherical triangles creates an $n$-gonal bipyramid, which represents the dihedral symmetry $D_{nh}$, order $4n$.

Different representations of the kaleidoscopic symmetry of certain small hosohedra

Symmetry (order $2n$)	Schönflies notation	$C_{nv}$	$C_{1v}$	$C_{2v}$	$C_{3v}$	$C_{4v}$	$C_{5v}$	$C_{6v}$
	Orbifold notation	$(*nn)$	$(*11)$	$(*22)$	$(*33)$	$(*44)$	$(*55)$	$(*66)$
	Coxeter diagram	File:CDel node.pngFile:CDel n.pngFile:CDel node.png	File:CDel node.png	File:CDel node.pngFile:CDel 2.pngFile:CDel node.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:CDel node.pngFile:CDel 5.pngFile:CDel node.png	File:CDel node.pngFile:CDel 6.pngFile:CDel node.png
		$[n]$	$[]$	$[2]$	$[3]$	$[4]$	$[5]$	$[6]$
$2n$-gonal hosohedron	Schläfli symbol	$\{2,2n\}$	$\{2,2\}$	$\{2,4\}$	$\{2,6\}$	$\{2,8\}$	$\{2,10\}$	$\{2,12\}$
	Alternately colored fundamental domains		File:Spherical digonal hosohedron2.png	File:Spherical square hosohedron2.png	File:Spherical hexagonal hosohedron2.png	File:Spherical octagonal hosohedron2.png	File:Spherical decagonal hosohedron2.png	File:Spherical dodecagonal hosohedron2.png

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:

File:Apeirogonal hosohedron.png

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

• Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc.,