Hosohedron
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} 
Schläfli symbol  {2,n} 
Wythoff symbol  n  2 2 
Coxeter–Dynkin diagrams  
Symmetry group  D_{nh}, [2,n], (*22n), order 4n 
Rotation group  D_{n}, [2,n]^{+}, (22n), order 2n 
Dual polyhedron  dihedron 
In geometry, an ngonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two vertices. A regular ngonal hosohedron has Schläfli symbol {2, n}.
Contents
Hosohedra as regular polyhedra
For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces may be found by:
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, represented as a tessellation of 4 spherical lunes on a sphere. 
1  2  3  4  5  6  7  8  9  10  11  12  ... 

{2,1} 
{2,2} 
{2,3} 
{2,4} 
{2,5} 
{2,6} 
{2,7} 
{2,8} 
{2,9} 
{2,10} 
{2,11} 
{2,12} 

Kalidescopic symmetry
The digonal faces of a 2nhosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: C_{nv}, [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 D_{nh}, order 4n.
Symmetry  C_{1v}  C_{2v}  C_{3v}  C_{4v}  C_{5v}  C_{6v} 

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.^{1}
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.
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 twodimensional hosotope {2} is a digon.
Etymology
The term “hosohedron” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (osos/hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”. ^{2}
See also
Wikimedia Commons has media related to Hosohedra. 
References
 ^ 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.
 Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0486614808
 Weisstein, Eric W., "Hosohedron", MathWorld.


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