# Hosohedron

(Redirected from Order-5-3 digonal honeycomb)
Short description: Spherical polyhedron composed of lunes
Set of regular n-gonal hosohedra Example regular hexagonal hosohedron on a sphere
Typeregular polyhedron or spherical tiling
Facesn digons
Edgesn
Vertices2
χ2
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 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.

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 {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

$\displaystyle{ N_2=\frac{4n}{2\times2+2n-2n}=n, }$

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.

## 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 $\displaystyle{ C_{1v} }$ $\displaystyle{ C_{2v} }$ $\displaystyle{ C_{3v} }$ $\displaystyle{ C_{4v} }$ $\displaystyle{ C_{5v} }$ $\displaystyle{ C_{6v} }$ $\displaystyle{ (*11) }$ $\displaystyle{ (*22) }$ $\displaystyle{ (*33) }$ $\displaystyle{ (*44) }$ $\displaystyle{ (*55) }$ $\displaystyle{ (*66) }$                $\displaystyle{ [\,\,] }$ $\displaystyle{  }$ $\displaystyle{  }$ $\displaystyle{  }$ $\displaystyle{  }$ $\displaystyle{  }$ $\displaystyle{ \{2,2\} }$ $\displaystyle{ \{2,4\} }$ $\displaystyle{ \{2,6\} }$ $\displaystyle{ \{2,8\} }$ $\displaystyle{ \{2,10\} }$ $\displaystyle{ \{2,12\} }$      ## Relationship with the Steinmetz solid

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.

## 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.