Hosohedron

In 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 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 polyhedrondihedron

A regular n-gonal hosohedron has Schläfli symbol {2, n}, with each spherical lune having internal angle 2π/n radians (360/n degrees).

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is {mn}, the number of polygonal faces may be found by:

$N_{2}={\frac {4n}{2m+2n-mn}}$

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 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 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, represented as a tessellation of 4 spherical lunes on a sphere.
Family of regular hosohedra (2 vertices)
n 2 3 4 5 6 7 8 9 10 11 12...
Image
{2,n} {2,2} {2,3} {2,4} {2,5} {2,6} {2,7} {2,8} {2,9} {2,10} {2,11} {2,12}
Coxeter

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.

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” was coined by H.S.M. Coxeter[dubious ], and possibly derives from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”.