# 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.
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

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

## 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

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.