# Hosohedron

Set of regular n-gonal hosohedra | |
---|---|

Example hexagonal hosohedron on a sphere | |

Type | Regular polyhedron or spherical tiling |

Faces |
n digons |

Edges |
n |

Vertices | 2 |

χ | 2 |

Vertex configuration |
2^{n} |

Wythoff symbol |
n | 2 2 |

Schläfli symbol |
{2,n} |

Coxeter diagram | |

Symmetry group |
D_{nh}, [2,n], (*22n), order 4n |

Rotation group |
D_{n}, [2,n]^{+}, (22n), order 2n |

Dual polyhedron | dihedron |

In 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.

A regular n-gonal hosohedron has Schläfli symbol {2, *n*}, with each spherical lune having internal angle 2π/*n* radians (360°/*n*).^{[1]}^{[2]}

## Hosohedra as regular polyhedra

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

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.

A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere. |
A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere. |

n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12... |
---|---|---|---|---|---|---|---|---|---|---|---|

Image | |||||||||||

{2,n} |
{2,2} | {2,3} | {2,4} | {2,5} | {2,6} | {2,7} | {2,8} | {2,9} | {2,10} | {2,11} | {2,12} |

Coxeter |

## Kaleidoscopic symmetry

The digonal (lune) faces of a 2*n*-hosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: C_{nv}, [n], (*nn), order 2*n*. 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 4*n*.

Symmetry | C_{1v}, [ ] |
C_{2v}, [2] |
C_{3v}, [3] |
C_{4v}, [4] |
C_{5v}, [5] |
C_{6v}, [6] |
---|---|---|---|---|---|---|

Hosohedron | {2,2} | {2,4} | {2,6} | {2,8} | {2,10} | {2,12} |

Fundamental domains |

## 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” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (*hosos*) “as many”, the idea being that a hosohedron can have “**as many** faces as desired”.^{[4]}

## See also

Wikimedia Commons has media related to .Hosohedra |

## References

- ↑ Coxeter,
*Regular polytopes*, p. 12 - ↑ Abstract Regular polytopes, p. 161
- ↑ Weisstein, Eric W. "Steinmetz Solid".
*MathWorld*. - ↑ Steven Schwartzman (1 January 1994).
*The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English*. MAA. pp. 108–109. ISBN 978-0-88385-511-9.

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