Hosohedron

Summary

Set of regular n-gonal hosohedra
Hexagonal Hosohedron.svg
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 diagramCDel node 1.pngCDel 2x.pngCDel node.pngCDel n.pngCDel node.png
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.

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/n radians (360/n degrees).[1][2]

Hosohedra as regular polyhedra

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.

Trigonal hosohedron.png
A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.
4hosohedron.svg
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 (Monogonal)
Henagonal hosohedron
Digonal hosohedron (Triangular)
Trigonal hosohedron
(Tetragonal)
Square hosohedron
Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedron Octagonal hosohedron Enneagonal hosohedron Decagonal hosohedron Hendecagonal hosohedron Dodecagonal hosohedron ... Apeirogonal hosohedron
Tiling image Spherical henagonal hosohedron.png Spherical digonal hosohedron.png Spherical trigonal hosohedron.png Spherical square hosohedron.png Spherical pentagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical heptagonal hosohedron.png Spherical octagonal hosohedron.png Spherical enneagonal hosohedron.png Spherical decagonal hosohedron.png Spherical hendecagonal hosohedron.png Spherical dodecagonal hosohedron.png ... Apeirogonal hosohedron.svg
Schläfli symbol {2,1} {2,2} {2,3} {2,4} {2,5} {2,6} {2,7} {2,8} {2,9} {2,10} {2,11} {2,12} ... {2,∞}
Coxeter diagram CDel node 1.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 7.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 8.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 9.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 10.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 11.pngCDel node.png CDel node 1.pngCDel 2x.pngCDel node.pngCDel 12.pngCDel node.png ... CDel node 1.pngCDel 2x.pngCDel node.pngCDel infin.pngCDel node.png
Faces and edges 1 2 3 4 5 6 7 8 9 10 11 12 ...
Vertices 2 ... 2
Vertex config. 2 2.2 23 24 25 26 27 28 29 210 211 212 ... 2

Kaleidoscopic symmetry

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

Bisecting each lune into two spherical triangles creates an n-gonal bipyramid, which represents the dihedral symmetry Dnh, order 4n.

Symmetry (order 2n) Cnv, [n] C1v, [ ] C2v, [2] C3v, [3] C4v, [4] C5v, [5] C6v, [6]
2n-gonal hosohedron Schläfli symbol {2,2n} {2,2} {2,4} {2,6} {2,8} {2,10} {2,12}
Image 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

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

Etymology

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

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.
  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) …

External links