thumb|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 -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees).
Hosohedra as regular polyhedra
For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :
:<math>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>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 . All these spherical lunes share two common vertices.
{| class="wikitable" width="320"
|160px<br />A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.
|160px<br />A regular tetragonal hosohedron, {2,4}, represented as a tessellation of 4 spherical lunes on a sphere.
|}
Kaleidoscopic symmetry
The <math>2n</math> digonal spherical lune faces of a <math>2n</math>-hosohedron, <math>\{2,2n\}</math>, represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry <math>C_{nv}</math>, <math>[n]</math>, <math>(*nn)</math>, order <math>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>n</math>-gonal bipyramid, which represents the dihedral symmetry <math>D_{nh}</math>, order <math>4n</math>.
{|class="wikitable" width=480
|+ Different representations of the kaleidoscopic symmetry of certain small hosohedra
|- align=center
! scope="row" rowspan=4 | Symmetry (order <math>2n</math>)
! scope="row" | Schönflies notation
! <math>C_{nv}</math>
| <math>C_{1v}</math>
| <math>C_{2v}</math>
| <math>C_{3v}</math>
| <math>C_{4v}</math>
| <math>C_{5v}</math>
| <math>C_{6v}</math>
|- align=center
! scope="row" | Orbifold notation
! <math>(*nn)</math>
| <math>(*11)</math>
| <math>(*22)</math>
| <math>(*33)</math>
| <math>(*44)</math>
| <math>(*55)</math>
| <math>(*66)</math>
|- align=center
! scope="row" rowspan=2 | Coxeter diagram
!
|
|
|
|
|
|
|- align=center
! <math>[n]</math>
| <math>[\,\,]</math>
| <math>[2]</math>
| <math>[3]</math>
| <math>[4]</math>
| <math>[5]</math>
| <math>[6]</math>
|-align=center
! scope="row" rowspan=2 | <math>2n</math>-gonal hosohedron
! scope="row" | Schläfli symbol
! <math>\{2,2n\}</math>
| <math>\{2,2\}</math>
| <math>\{2,4\}</math>
| <math>\{2,6\}</math>
| <math>\{2,8\}</math>
| <math>\{2,10\}</math>
| <math>\{2,12\}</math>
|-
! scope="row" colspan=2 | Alternately colored fundamental domains
|80px
|80px
|80px
|80px
|80px
|80px
|}
Relationship with the Steinmetz solid
The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.
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:
:frameless
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”. It was introduced by Vito Caravelli in the eighteenth century.
See also
- Polyhedron
- Polytope
References
- Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc.,