Set of regularn-gonal hosohedra
Exampleregular hexagonal hosohedron on a sphere
Type	regularpolyhedron orspherical tiling
Faces	ndigons
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) order4n
Rotation group	Dn [2,n]+ (22n) order2n
Dual polyhedron	regularn-gonaldihedron

Inspherical geometry, ann-gonalhosohedron is atessellation oflunes on aspherical surface, such that each lune shares the same twopolar opposite vertices.

Aregularn-gonal hosohedron hasSchläfli symbol{2,n}, with eachspherical lune havinginternal angle⁠2π/n⁠radians (⁠360/n⁠ degrees).^[1][2]

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

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

ThePlatonic solids known to antiquity are the only integer solutions form ≥ 3 andn ≥ 3. The restrictionm ≥ 3 enforces that the polygonal faces must have at least three sides.

When considering polyhedra as aspherical tiling, this restriction may be relaxed, sincedigons (2-gons) can be represented asspherical lunes, having non-zeroarea.

Allowingm = 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 asn abutting lunes, with interior angles of⁠2π/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∞

The${\displaystyle 2n}$ digonalspherical lune faces of a${\displaystyle 2n}$-hosohedron,${\displaystyle \{2,2n\}}$, represent the fundamental domains ofdihedral symmetry in three dimensions: the cyclic symmetry${\displaystyle C_{nv}}$,${\displaystyle [n]}$,${\displaystyle (\ast nn)}$, 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${\displaystyle n}$-gonalbipyramid, which represents thedihedral 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	${\displaystyle (\ast nn)}$	${\displaystyle (\ast 11)}$	${\displaystyle (\ast 22)}$	${\displaystyle (\ast 33)}$	${\displaystyle (\ast 44)}$	${\displaystyle (\ast 55)}$	${\displaystyle (\ast 66)}$
	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

The tetragonal hosohedron is topologically equivalent to thebicylinder Steinmetz solid, the intersection of two cylinders at right-angles.^[3]

Thedual of the n-gonal hosohedron {2, n} is then-gonaldihedron, {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 atruncated variation. The truncatedn-gonal hosohedron is the n-gonalprism.

In the limit, the hosohedron becomes anapeirogonal hosohedron as a 2-dimensional tessellation:

Multidimensional analogues in general are calledhosotopes. A regular hosotope withSchläfli symbol {2,p,...,q} has two vertices, each with avertex figure {p,...,q}.

Thetwo-dimensional hosotope, {2}, is adigon.

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]

Wikimedia Commons has media related toHosohedra.

• Polyhedron
• Polytope

• 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

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

• Polyhedra
• Tessellation
• Regular polyhedra

• Articles with short description
• Short description is different from Wikidata
• Commons category link is on Wikidata