| Set of regular q-gonal hosohedrons | |
|---|---|
 Example hexagonal hosohedron on a sphere |
|
| Type | Regular polyhedron or spherical tiling |
| Faces | q digons |
| Edges | q |
| Vertices | 2 |
| Schläfli symbol | {2,q} |
| Vertex configuration | 2q |
| Coxeter–Dynkin diagram |     |
| Wythoff symbol | q | 2 2 |
| Symmetry group | Dihedral (Dqh) |
| 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 vertices. A regular n-gonal hosohedron has Schläfli symbol {2, n}.
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Hosohedrons as regular polyhedrons
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 polyhedrons as a spherical tiling, 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.
 A regular trigonal hosohedron, 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. |
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.
Hosotopes
Multidimensional analogues in general are called hosotopes, with Schläfli symbol {2,...,2,q}. A hosotope has two vertices.
The two-dimensional hosotope {2} is a digon.
Etymology
The prefix “hoso-” was invented by H.S.M. Coxeter, and possibly derives from the English “hose”.
See also
References
- Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
- Weisstein, Eric W., "Hosohedron" from MathWorld.
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