Hosohedron

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Set of regular n-gonal hosohedra
Hexagonal Hosohedron.svg
Example hexagonal hosohedron on a sphere
Type Regular polyhedron or spherical tiling
Faces n digons
Edges n
Vertices 2
χ 2
Vertex configuration 2n
Schläfli symbol {2,n}
Wythoff symbol n | 2 2
Coxeter–Dynkin diagrams CDel node.pngCDel n.pngCDel node.pngCDel 2.pngCDel node 1.png
Symmetry group Dnh, [2,n], (*22n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron dihedron
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[edit]

For a regular polyhedron whose Schläfli symbol is {mn}, 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.

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, 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 ...
CDel node 1.pngCDel 2.pngCDel node.png
{2,1}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
{2,2}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
{2,3}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
{2,4}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
{2,5}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
{2,6}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 7.pngCDel node.png
{2,7}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 8.pngCDel node.png
{2,8}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 9.pngCDel node.png
{2,9}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 1x.pngCDel 0x.pngCDel node.png
{2,10}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 1x.pngCDel 1x.pngCDel node.png
{2,11}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 1x.pngCDel 2x.pngCDel node.png
{2,12}
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

Kaleidoscopic symmetry[edit]

The digonal 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 C3v C4v C5v C6v
Hosohedron {2,2} {2,4} {2,6} {2,8} {2,10} {2,12}
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[edit]

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.[1]

Derivative polyhedra[edit]

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[edit]

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[edit]

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”. [2]

See also[edit]

References[edit]