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).

For a regular polyhedron whose Schläfli symbol is {m, n}, 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 of⁠2π/n⁠. All these spherical lunes share two common vertices.

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 ${\displaystyle 4n}$.

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

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.

In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

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.

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.

• Polyhedron
• Polytope

• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0
• Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc.,ISBN 0-486-61480-8

• Weisstein, Eric W. "Hosohedron". MathWorld.