Set of regular n-gonal hosohedra
Example regular hexagonal hosohedron on a sphere
Typeregular polyhedron or spherical tiling
Facesn digons
Edgesn
Vertices2
Euler char.2
Vertex configuration2n
Wythoff symboln | 2 2
Schläfli symbol{2,n}
Coxeter diagram
Symmetry groupDnh
[2,n]
(*22n)

order 4n
Rotation groupDn
[2,n]+
(22n)

order 2n
Dual polyhedronregular n-gonal dihedron

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).[1][2]

## Hosohedra as regular polyhedra

 Further information: List of regular polytopes and compounds § Spherical 2

For a regular polyhedron whose Schläfli symbol is {mn}, 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 ${\displaystyle 4n}$.

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

## 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]