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