Hosohedron

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A tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.
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A tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.
A colored hexagonal hosohedron, represented as a tessellation of 6 spherical lune faces on a sphere. {2,6}
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A colored hexagonal hosohedron, represented as a tessellation of 6 spherical lune faces on a sphere. {2,6}
This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.
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This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.

An n-gonal hosohedron is a tesselation of lunes on a spherical surface, such that each lune shares the same two vertices. Its Schläfli symbol is {2, n}.

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[edit] Hosohedrons as regular polyhedrons

For a regular polyhedron whose Schläfli symbol is {m, n}, 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 polyhedrons as regular tessellations on a spherical surface, this restriction may be relaxed, since digons can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedrons, which are the hosohedrons. 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.

[edit] Derivative polyhedrons

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 polyhedrons to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism.

[edit] Multidimensional analogues

The 4-dimensional analogue is called a hosochoron (plural: hosochora). For example, {2,3,3} could be called a tetrahedral hosochoron.[citation needed]

Multidimensional analogues in general are called hosotopes. A hosotope has two facets. The two-dimensional hosotope {2} is a digon.

[edit] Etymology

The prefix “hoso-” was invented by H.S.M. Coxeter, and possibly derives from the English “hose”.

[edit] See also

[edit] References

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