Hosohedron

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is : :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 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 makes :N_2=\frac{4n}{2\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.

Kaleidoscopic symmetry

The 2n digonal spherical lune faces of a 2n-hosohedron, {2,2n}, 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.

Relationship with the Steinmetz solid

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

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: :[[File:Apeirogonal hosohedron.png|frameless]]

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” appears to derive from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”. It was introduced by Vito Caravelli in the eighteenth century.

References

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

References

1. Coxeter, ''Regular polytopes'', p. 12
2. Abstract Regular polytopes, p. 161
3. "Steinmetz Solid".
4. Steven Schwartzman. (1 January 1994). "The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English". MAA.
5. Coxeter, H.S.M.. (1974). "Regular Complex Polytopes". Cambridge University Press.