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 $2n$ with each spherical lune having internal angle ${2,n}$radians ($n | 2 2$ 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
 * $$N_2=\frac{4n}{2\times2+2n-2n}=n,$$

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 $n$. All these spherical lunes share two common vertices.

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 $$4n$$.

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:
 * Apeirogonal hosohedron.png

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.