In geometry, an ngonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
Set of regular ngonal hosohedra  

Example hexagonal hosohedron on a sphere


Type  Regular polyhedron or spherical tiling 
Faces  n digons 
Edges  n 
Vertices  2 
χ  2 
Vertex configuration  2^{n} 
Wythoff symbol  n  2 2 
Schläfli symbol  {2,n} 
Coxeter diagram  
Symmetry group  D_{nh}, [2,n], (*22n), order 4n 
Rotation group  D_{n}, [2,n]^{+}, (22n), order 2n 
Dual polyhedron  dihedron 
A regular ngonal hosohedron has Schläfli symbol {2, n}, with each spherical lune having internal angle 2π/n radians (360/n degrees).[1][2]
Hosohedra as regular polyhedra
Further information: List_of_regular_polytopes_and_compounds § Spherical_2
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+2nmn}}} N_{2}={\frac {4n}{2m+2nmn}}
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 (2gons) can be represented as spherical lunes, having nonzero area. Allowing m = 2 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 lunes share two common vertices.
A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere. 
A regular tetragonal hosohedron, {2,4}, represented as a tessellation of 4 spherical lunes on a sphere. 
Family of regular hosohedra (2 vertices) n 2 3 4 5 6 7 8 9 10 11 12...
n  2  3  4  5  6  7  8  9  10  11  12... 

Image  
{2,n}  {2,2}  {2,3}  {2,4}  {2,5}  {2,6}  {2,7}  {2,8}  {2,9}  {2,10}  {2,11}  {2,12} 
Coxeter 
Kaleidoscopic symmetry
The digonal (lune) faces of a 2nhosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: Cnv, [n], (*nn), order 2n. The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n.
Symmetry  C_{1v}, [ ]  C_{2v}, [2]  C_{3v}, [3]  C_{4v}, [4]  C_{5v}, [5]  C_{6v}, [6] 

Hosohedron  {2,2}  {2,4}  {2,6}  {2,8}  {2,10}  {2,12} 
Fundamental domains 
Relationship with the Steinmetz solid
The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at rightangles.[3]
Derivative polyhedra
The dual of the ngonal hosohedron {2, n} is the ngonal dihedron, {n, 2}. The polyhedron {2,2} is selfdual, 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 ngonal hosohedron is the ngonal prism.
Apeirogonal hosohedron
In the limit the hosohedron becomes an apeirogonal hosohedron as a 2dimensional tessellation:
Hosotopes
Further information: List_of_regular_polytopes_and_compounds § Spherical_3
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 twodimensional hosotope, {2}, is a digon.
Etymology
The term “hosohedron” was coined by H.S.M. Coxeter[dubious – discuss], and possibly derives from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”.[4]
See also
Polyhedron
Polytope
References
Coxeter, Regular polytopes, p. 12
Abstract Regular polytopes, p. 161
Weisstein, Eric W. "Steinmetz Solid". MathWorld.
Steven Schwartzman (1 January 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. pp. 108–109. ISBN 9780883855119.
McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0521814960
Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0486614808
External links
Weisstein, Eric W. "Hosohedron". MathWorld.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World  Scientific Library
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