Stratonovich integral

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File:BeachBall.jpg

This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schläfli symbol {2, n}.

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces may be found by:

$N_2=\frac{4n}{2m+2n-mn}$

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.

File:Trigonal hosohedron.png A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.

File:4hosohedron.svg A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.

Family of regular hosohedra

1	2	3	4	5	6	7	8	9	10	11	12	...
Template:CDD {2,1}	Template:CDD {2,2}	Template:CDD {2,3}	Template:CDD {2,4}	Template:CDD {2,5}	Template:CDD {2,6}	Template:CDD {2,7}	Template:CDD {2,8}	Template:CDD {2,9}	Template:CDD {2,10}	Template:CDD {2,11}	Template:CDD {2,12}
File:Spherical henagonal hosohedron.png	File:Spherical digonal hosohedron.png	File:Spherical trigonal hosohedron.png	File:Spherical square hosohedron.png	File:Spherical pentagonal hosohedron.png	File:Spherical hexagonal hosohedron.png	File:Spherical heptagonal hosohedron.png	File:Spherical octagonal hosohedron.png	File:Spherical enneagonal hosohedron.png	File:Spherical decagonal hosohedron.png	File:Spherical hendecagonal hosohedron.png	File:Spherical dodecagonal hosohedron.png

Kalidescopic symmetry

The digonal faces of a 2n-hosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: C_nv, [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 D_nh, order 4n.

Symmetry	C1v	C2v	C3v	C4v	C5v	C6v
Hosohedron	{2,2}	{2,4}	{2,6}	{2,8}	{2,10}	{2,12}
Fundamental domains	File:Spherical digonal hosohedron2.png	File:Spherical square hosohedron2.png	File:Spherical hexagonal hosohedron2.png	File:Spherical octagonal hosohedron2.png	File:Spherical decagonal hosohedron2.png	File: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.

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.

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” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (osos/hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”.

See also

• Polyhedron
• Polytope

References

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• Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
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