Hosohedron

In 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 hexagonal hosohedron on a sphere
TypeRegular polyhedron or spherical tiling
Facesn digons
Edgesn
Vertices2
χ2
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 polyhedrondihedron
This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

A regular n-gonal 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 :

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 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)
n23456789101112...
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 2n-hosohedron, {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 C1v, [ ] C2v, [2] C3v, [3] C4v, [4] C5v, [5] C6v, [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 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:

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 two-dimensional hosotope, {2}, is a digon.

Etymology

The term “hosohedron” was coined by H.S.M. Coxeter, 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

References
  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.
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