Hosohedron

Inspherical geometry,ann-gonalhosohedronis atessellationofluneson aspherical surface,such that each lune shares the same twopolar oppositevertices.

Set of regularn-gonal hosohedra
Exampleregularhexagonal hosohedron on a sphere
Type	regularpolyhedronorspherical tiling
Faces	ndigons
Edges	n
Vertices	2
Euler char.	2
Vertex configuration	2n
Wythoff symbol	n| 2 2
Schläfli symbol	{2,n}
Coxeter diagram
Symmetry group	Dnh [2,n] (*22n) order4n
Rotation group	Dn [2,n]+ (22n) order2n
Dual polyhedron	regularn-gonaldihedron

Thisbeach ballwould be a hosohedron with 6spherical lunefaces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.

Aregularn-gonal hosohedron hasSchläfli symbol{2,n},with eachspherical lunehavinginternal angle⁠2π/n⁠radians(⁠360/n⁠degrees).

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:

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

ThePlatonic solidsknown to antiquity are the only integer solutions form≥ 3 andn≥ 3. The restrictionm≥ 3 enforces that the polygonal faces must have at least three sides.

When considering polyhedra as aspherical tiling,this restriction may be relaxed, sincedigons(2-gons) can be represented asspherical lunes,having non-zeroarea.

Allowingm= 2 makes

$N_{2}={\frac {4n}{2\times 2+2n-2n}}=n,$

and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2,n} is represented asnabutting lunes, with interior angles of⁠2π/n⁠.All these spherical 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 · *n22 symmetry mutations of regular hosohedral tilings:nn

Space	Spherical						Euclidean
Tiling name	Henagonal hosohedron	Digonal hosohedron	Trigonal hosohedron	Square hosohedron	Pentagonal hosohedron	...	Apeirogonal hosohedron
Tiling image						...
Schläfli symbol	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	...	{2,∞}
Coxeter diagram						...
Faces and edges	1	2	3	4	5	...	∞
Vertices	2	2	2	2	2	...	2
Vertex config.	2	2.2	23	24	25	...	2∞

Kaleidoscopic symmetry

The$2n$digonalspherical lunefaces of a$2n$-hosohedron,$\{2,2n\}$,represent the fundamental domains ofdihedral symmetry in three dimensions:the cyclic symmetry$C_{nv}$,$[n]$,$(*nn)$,order$2n$.The reflection domains can be shown by alternately colored lunes as mirror images.

Bisecting each lune into two spherical triangles creates an$n$-gonalbipyramid,which represents thedihedral symmetry$D_{nh}$,order$4n$.

Different representations of the kaleidoscopic symmetry of certain small hosohedra

Symmetry (order$2n$)	Schönflies notation	$C_{nv}$	$C_{1v}$	$C_{2v}$	$C_{3v}$	$C_{4v}$	$C_{5v}$	$C_{6v}$
	Orbifold notation	$(*nn)$	$(*11)$	$(*22)$	$(*33)$	$(*44)$	$(*55)$	$(*66)$
	Coxeter diagram
		$[n]$	$[\,\,]$	$[2]$	$[3]$	$[4]$	$[5]$	$[6]$
$2n$-gonal hosohedron	Schläfli symbol	$\{2,2n\}$	$\{2,2\}$	$\{2,4\}$	$\{2,6\}$	$\{2,8\}$	$\{2,10\}$	$\{2,12\}$
	Alternately colored fundamental domains

Relationship with the Steinmetz solid

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

Derivative polyhedra

Thedualof the n-gonal hosohedron {2,n} is then-gonaldihedron,{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 atruncatedvariation. The truncatedn-gonal hosohedron is the n-gonalprism.

Apeirogonal hosohedron

In the limit, the hosohedron becomes anapeirogonal hosohedronas a 2-dimensional tessellation:

Hosotopes

Further information:List of regular polytopes and compounds § Spherical 3

Multidimensionalanalogues in general are calledhosotopes.A regular hosotope withSchläfli symbol{2,p,...,q} has two vertices, each with avertex figure{p,...,q}.

Thetwo-dimensional hosotope,{2}, is adigon.

Etymology

The term “hosohedron” appears to derive from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as manyfaces as desired”.It was introduced by Vito Caravelli in the eighteenth century.

See also

• Polyhedron
• Polytope

References

• McMullen, Peter;Schulte, Egon (December 2002),Abstract Regular Polytopes(1st ed.),Cambridge University Press,ISBN0-521-81496-0
• Coxeter, H.S.M,Regular Polytopes(third edition), Dover Publications Inc.,ISBN0-486-61480-8

External links

• Weisstein, Eric W."Hosohedron".MathWorld.