# Hosohedron

Last updated
Set of regular n-gonal hosohedra
Example regular hexagonal hosohedron on a sphere
Typeregular polyhedron or spherical tiling
Faces n digons
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)

order 4n
Rotation group Dn
[2,n]+
(22n)

order 2n
Dual polyhedron regular n-gonal dihedron

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.

## Contents

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 {m, n}, the number of polygonal faces is :

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

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 makes

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

and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2, n} is represented as n abutting lunes, with interior angles of 2π/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
SpaceSphericalEuclidean
Tiling name(Monogonal)
Henagonal hosohedron
Digonal hosohedron (Triangular)
Trigonal hosohedron
(Tetragonal)
Square hosohedron
Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedronOctagonal hosohedronEnneagonal hosohedronDecagonal hosohedronHendecagonal hosohedronDodecagonal hosohedron... Apeirogonal hosohedron
Tiling image ...
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 ...
Faces and edges123456789101112...
Vertices2...2
Vertex config. 22.223242526272829210211212...2

## Kaleidoscopic symmetry

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

Bisecting each lune into two spherical triangles creates an ${\displaystyle n}$-gonal bipyramid, which represents the dihedral symmetry ${\displaystyle D_{nh}}$, order ${\displaystyle 4n}$.

Symmetry (order ${\displaystyle 2n}$) Schönflies notation ${\displaystyle C_{nv}}$ Orbifold notation ${\displaystyle (*nn)}$ Coxeter diagram ${\displaystyle C_{1v}}$ ${\displaystyle C_{2v}}$ ${\displaystyle C_{3v}}$ ${\displaystyle C_{4v}}$ ${\displaystyle C_{5v}}$ ${\displaystyle C_{6v}}$ ${\displaystyle (*11)}$ ${\displaystyle (*22)}$ ${\displaystyle (*33)}$ ${\displaystyle (*44)}$ ${\displaystyle (*55)}$ ${\displaystyle (*66)}$ ${\displaystyle [\,\,]}$ ${\displaystyle [2]}$ ${\displaystyle [3]}$ ${\displaystyle [4]}$ ${\displaystyle [5]}$ ${\displaystyle [6]}$ ${\displaystyle \{2,2\}}$ ${\displaystyle \{2,4\}}$ ${\displaystyle \{2,6\}}$ ${\displaystyle \{2,8\}}$ ${\displaystyle \{2,10\}}$ ${\displaystyle \{2,12\}}$

## 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:

## 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]

## Related Research Articles

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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

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In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

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## 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). . 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) …