Hosohedron

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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.

Set of regular n-gonal hosohedra
Example regular hexagonal hosohedron on a sphere
Typeregular polyhedron or spherical tiling
Facesn digons
Edgesn
Vertices2
Euler char.2
Vertex configuration2n
Wythoff symboln | 2 2
Schläfli symbol{2,n}
Coxeter diagram
Symmetry groupDnh
[2,n]
(*22n)

order 4n
Rotation groupDn
[2,n]+
(22n)

order 2n
Dual polyhedronregular n-gonal dihedron
This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.

A regular n-gonal hosohedron has Schläfli symbol {2,n}, with each spherical lune having internal angle 2π/nradians (360/n degrees).[1][2]

Hosohedra as regular polyhedra

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For a regular polyhedron whose Schläfli symbol is {mn}, 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 2π/n. All these spherical 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 · *n22 symmetry mutations of regular hosohedral tilings: nn
Space Spherical Euclidean
Tiling
name
Henagonal
hosohedron
Digonal
hosohedron
Trigonal
hosohedron
Square
hosohedron
Pentagonal
hosohedron
... Apeirogonal
hosohedron
Tiling
image
          ...  
Schläfli
symbol
{2,1} {2,2} {2,3} {2,4} {2,5} ... {2,∞}
Coxeter
diagram
                            ...      
Faces and
edges
1 2 3 4 5 ...
Vertices 2 2 2 2 2 ... 2
Vertex
config.
2 2.2 23 24 25 ... 2

Kaleidoscopic symmetry

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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  .

Different representations of the kaleidoscopic symmetry of certain small hosohedra
Symmetry (order  ) Schönflies notation              
Orbifold notation              
Coxeter diagram                          
             
 -gonal hosohedron Schläfli symbol              
Alternately colored fundamental domains            

Relationship with the Steinmetz solid

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The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.[3]

Derivative polyhedra

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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

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In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

 

Hosotopes

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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

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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”.[4] It was introduced by Vito Caravelli in the eighteenth century.[5]

See also

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References

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  1. ^ Coxeter, Regular polytopes, p. 12
  2. ^ Abstract Regular polytopes, p. 161
  3. ^ Weisstein, Eric W. "Steinmetz Solid". MathWorld.
  4. ^ Steven Schwartzman (1 January 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. pp. 108–109. ISBN 978-0-88385-511-9.
  5. ^ Coxeter, H.S.M. (1974). Regular Complex Polytopes. London: Cambridge University Press. p. 20. ISBN 0-521-20125-X. The hosohedron {2,p} (in a slightly distorted form) was named by Vito Caravelli (1724–1800) …
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