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).

[1][2] For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is : The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3.

Allowing m = 2 makes 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.

The reflection domains can be shown by alternately colored lunes as mirror images.

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

A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation.

A regular hosotope with Schläfli symbol {2,p,...,q} has two vertices, each with a vertex figure {p,...,q}.