Hosohedron

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.

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
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	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.

A regular n-gonal hosohedron has Schläfli symbol {2,n}, with each spherical lune having internal angle ⁠2π/n⁠radians (⁠360/n⁠ degrees).^[1][2]

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 :

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

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

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

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.

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 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						...
Schläfli symbol	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	...	{2,∞}
Coxeter diagram						...
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 ${\displaystyle 2n}$ digonal spherical lune faces of a ${\displaystyle 2n}$-hosohedron, ${\displaystyle \{2,2n\}}$, represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry ${\displaystyle C_{nv}}$, ${\displaystyle [n]}$, ${\displaystyle (*nn)}$, order ${\displaystyle 2n}$. The reflection domains can be shown by alternately colored lunes as mirror images.

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

Different representations of the kaleidoscopic symmetry of certain small hosohedra

Symmetry (order ${\displaystyle 2n}$)	Schönflies notation	${\displaystyle C_{nv}}$	${\displaystyle C_{1v}}$	${\displaystyle C_{2v}}$	${\displaystyle C_{3v}}$	${\displaystyle C_{4v}}$	${\displaystyle C_{5v}}$	${\displaystyle C_{6v}}$
	Orbifold notation	${\displaystyle (*nn)}$	${\displaystyle (*11)}$	${\displaystyle (*22)}$	${\displaystyle (*33)}$	${\displaystyle (*44)}$	${\displaystyle (*55)}$	${\displaystyle (*66)}$
	Coxeter diagram
		${\displaystyle [n]}$	${\displaystyle [\,\,]}$	${\displaystyle [2]}$	${\displaystyle [3]}$	${\displaystyle [4]}$	${\displaystyle [5]}$	${\displaystyle [6]}$
${\displaystyle 2n}$-gonal hosohedron	Schläfli symbol	${\displaystyle \{2,2n\}}$	${\displaystyle \{2,2\}}$	${\displaystyle \{2,4\}}$	${\displaystyle \{2,6\}}$	${\displaystyle \{2,8\}}$	${\displaystyle \{2,10\}}$	${\displaystyle \{2,12\}}$
	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.^[3]

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”.^[4] It was introduced by Vito Caravelli in the eighteenth century.^[5]

See also

• Polyhedron
• Polytope

References

1. Coxeter, Regular polytopes, p. 12
2. Abstract Regular polytopes, p. 161
3. Weisstein, Eric W. "Steinmetz Solid". MathWorld.
4. Steven Schwartzman (1 January 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. pp. 108–109. ISBN 978-0-88385-511-9.
5. Coxeter, H.S.M. (1974). Regular Complex Polytopes. London: Cambridge University Press. p. 20. ISBN 0-521-20125-X. The hosohedron {2,p} (in a slightly distorted form) was named by Vito Caravelli (1724–1800) …

• 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.