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).

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :

N 2 = 4 n 2 m + 2 n − m n . {\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

N 2 = 4 n 2 × 2 + 2 n − 2 n = n , {\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 of ⁠2π/n⁠. All these spherical lunes share two common vertices.

Kaleidoscopic symmetry The 2 n {\displaystyle 2n} digonal spherical lune faces of a 2 n {\displaystyle 2n} -hosohedron, { 2 , 2 n } {\displaystyle \{2,2n\}} , represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry C n v {\displaystyle C_{nv}} , [ n ] {\displaystyle [n]} , ( ∗ n n ) {\displaystyle (*nn)} , order 2 n {\displaystyle 2n} . The reflection domains can be shown by alternately colored lunes as mirror images. Bisecting each lune into two spherical triangles creates an n {\displaystyle n} -gonal bipyramid, which represents the dihedral symmetry D n h {\displaystyle D_{nh}} , order 4 n {\displaystyle 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:

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.

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