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). Hosohedra as regular polyhedra 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. 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 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. Kaleidoscopic symmetry 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 . 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: 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. See also Polyhedron Polytope References Coxeter, H.S.M, Regular Polytopes (third