Hosohedron

In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees).

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is , the number of polygonal faces is :
:$N_2=\frac.$

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
:$N_2=\frac=n,$
and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron 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, $\$, represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry $C_$, /math>, $(*nn)$, 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 $D_$, 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 is the ''n''-gonal dihedron, . The polyhedron 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:
:

Hosotopes

Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has two vertices, each with a vertex figure .

The two-dimensional hosotope, , 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.

See also

* Polyhedron
* Polytope

References

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

External links

*

{{Tessellation

Polyhedra
Tessellation
Regular polyhedra