In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees).
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 âÂÂ¥ 3 and n âÂÂ¥ 3. The restriction m âÂÂ¥ 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
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 . All these spherical lunes share two common vertices.
The digonal spherical lune faces of a -hosohedron, , represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry , , , order . The reflection domains can be shown by alternately colored lunes as mirror images.
Bisecting each lune into two spherical triangles creates an -gonal bipyramid, which represents the dihedral symmetry , order .
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 âÂÂhosohedronâ appears to derive from the Greek á½ ÃÂÿà(hosos) âÂÂas manyâÂÂ, the idea being that a hosohedron can have âÂÂas many faces as desiredâÂÂ. It was introduced by Vito Caravelli in the eighteenth century.