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Short description: Spherical polyhedron composed of lunes
Set of regular n-gonal hosohedra
Hexagonal Hosohedron.svg
Example regular hexagonal hosohedron on a sphere
Typeregular polyhedron or spherical tiling
Facesn digons
Vertex configuration2n
Wythoff symboln | 2 2
Schläfli symbol{2,n}
Coxeter diagramCDel node 1.pngCDel 2x.pngCDel node.pngCDel n.pngCDel node.png
Symmetry groupDnh

order 4n
Rotation groupDn

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.

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).[1][2]

Hosohedra as regular polyhedra

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

[math]\displaystyle{ N_2=\frac{4n}{2m+2n-mn}. }[/math]

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

[math]\displaystyle{ N_2=\frac{4n}{2\times2+2n-2n}=n, }[/math]

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.

Trigonal hosohedron.png
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.

Kaleidoscopic symmetry

The [math]\displaystyle{ 2n }[/math] digonal spherical lune faces of a [math]\displaystyle{ 2n }[/math]-hosohedron, [math]\displaystyle{ \{2,2n\} }[/math], represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry [math]\displaystyle{ C_{nv} }[/math], [math]\displaystyle{ [n] }[/math], [math]\displaystyle{ (*nn) }[/math], order [math]\displaystyle{ 2n }[/math]. The reflection domains can be shown by alternately colored lunes as mirror images.

Bisecting each lune into two spherical triangles creates an [math]\displaystyle{ n }[/math]-gonal bipyramid, which represents the dihedral symmetry [math]\displaystyle{ D_{nh} }[/math], order [math]\displaystyle{ 4n }[/math].

Different representations of the kaleidoscopic symmetry of certain small hosohedra
Symmetry (order [math]\displaystyle{ 2n }[/math]) Schönflies notation [math]\displaystyle{ C_{nv} }[/math] [math]\displaystyle{ C_{1v} }[/math] [math]\displaystyle{ C_{2v} }[/math] [math]\displaystyle{ C_{3v} }[/math] [math]\displaystyle{ C_{4v} }[/math] [math]\displaystyle{ C_{5v} }[/math] [math]\displaystyle{ C_{6v} }[/math]
Orbifold notation [math]\displaystyle{ (*nn) }[/math] [math]\displaystyle{ (*11) }[/math] [math]\displaystyle{ (*22) }[/math] [math]\displaystyle{ (*33) }[/math] [math]\displaystyle{ (*44) }[/math] [math]\displaystyle{ (*55) }[/math] [math]\displaystyle{ (*66) }[/math]
Coxeter diagram CDel node.pngCDel n.pngCDel node.png CDel node.png CDel node.pngCDel 2.pngCDel node.png CDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node.png
[math]\displaystyle{ [n] }[/math] [math]\displaystyle{ [\,\,] }[/math] [math]\displaystyle{ [2] }[/math] [math]\displaystyle{ [3] }[/math] [math]\displaystyle{ [4] }[/math] [math]\displaystyle{ [5] }[/math] [math]\displaystyle{ [6] }[/math]
[math]\displaystyle{ 2n }[/math]-gonal hosohedron Schläfli symbol [math]\displaystyle{ \{2,2n\} }[/math] [math]\displaystyle{ \{2,2\} }[/math] [math]\displaystyle{ \{2,4\} }[/math] [math]\displaystyle{ \{2,6\} }[/math] [math]\displaystyle{ \{2,8\} }[/math] [math]\displaystyle{ \{2,10\} }[/math] [math]\displaystyle{ \{2,12\} }[/math]
Alternately colored fundamental domains Spherical digonal hosohedron2.png Spherical square hosohedron2.png Spherical hexagonal hosohedron2.png Spherical octagonal hosohedron2.png Spherical decagonal hosohedron2.png Spherical dodecagonal hosohedron2.png

Relationship with the Steinmetz solid

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

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:

Apeirogonal hosohedron.png


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.


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


  1. Coxeter, Regular polytopes, p. 12
  2. Abstract Regular polytopes, p. 161
  3. Weisstein, Eric W.. "Steinmetz Solid". http://mathworld.wolfram.com/SteinmetzSolid.html. 
  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. https://archive.org/details/wordsofmathemati0000schw. 
  5. Coxeter, H.S.M. (1974). Regular Complex Polytopes. London: Cambridge University Press. pp. 20. ISBN 0-521-20125-X. "The hosohedron {2,p} (in a slightly distorted form) was named by Vito Caravelli (1724–1800) …" 

External links