Hosohedron
An n-gonal hosohedron is a degenerate case of a regular polyhedron, in which n digons (two-sided polygons) meet at each vertex. Its Schläfli symbol is {2,n}.
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 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 vertecies.
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 trunctated n-gonal hosohedron is the n-gonal prism.
Multidimensional Analogues
The 4-dimensional analogues are called hosochorons. For example, {3,3,2} is a tetrahedral hosochoron.
Multidimensional analogues in general are called hosotopes. In these, the last element in the Schläfli symbol is a 2. 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
- Wolfram Research (http://mathworld.wolfram.com/Hosohedron.html) Retrieved Jul 7, 2005.