Hosohedron

 ( Polyhedra ) 

In spherical geometry, an Polygon hosohedron is a tessellation of spherical lune on a Sphere, such that each lune shares the same two Antipodal point vertices.

A Regular polygon -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle ( degrees).Coxeter, Regular polytopes, p. 12Abstract Regular polytopes, p. 161

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Hosohedra as regular polyhedra For a regular polyhedron whose Schläfli symbol is { m,  n}, the number of polygonal faces is :

$N\_2=\backslash frac\{4n\}\{2m+2n-mn\}.$

The 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 (2-gons) can be represented as , having non-zero area.

Allowing m = 2 makes

$N\_2=\backslash frac\{4n\}\{2\backslash times2+2n-2n\}=n,$

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 . 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.

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Kaleidoscopic symmetry The $2n$ digonal spherical lune faces of a $2n$-hosohedron, $\backslash \{2,2n\backslash \}$, represent the fundamental domains of dihedral 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 $n$-gonal bipyramid, which represents the dihedral symmetry $D\_\{nh\}$, order $4n$.

+ Different representations of the kaleidoscopic symmetry of certain small hosohedra

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

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Derivative polyhedra The dual polyhedron 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.

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Apeirogonal hosohedron In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

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Hosotopes dimension 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.

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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”.

Steven Schwartzman (1994). 9780883855119, MAA. . ISBN 9780883855119

It was introduced by Vito Caravelli in the eighteenth century.

Coxeter, H.S.M. (1974). 052120125X, Cambridge University Press. 052120125X

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See also

• Polyhedron
• Polytope

• Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc.,

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External links

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