Hosohedron

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :

${\displaystyle N_{2}={\frac {4n}{2m+2n-mn}}.}$

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

${\displaystyle N_{2}={\frac {4n}{2\times 2+2n-2n}}=n,}$

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 2π/n. All these spherical lunes share two common vertices.

A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.

A regular tetragonal hosohedron, {2,4}, represented as a tessellation of 4 spherical lunes on a sphere.

Family of regular hosohedra · *n22 symmetry mutations of regular hosohedral tilings: nn

Space	Spherical													Euclidean
Tiling name	(Monogonal) Henagonal hosohedron	Digonal hosohedron	(Triangular) Trigonal hosohedron	(Tetragonal) Square hosohedron	Pentagonal hosohedron	Hexagonal hosohedron	Heptagonal hosohedron	Octagonal hosohedron	Enneagonal hosohedron	Decagonal hosohedron	Hendecagonal hosohedron	Dodecagonal hosohedron	...	Apeirogonal hosohedron
Tiling image													...
Schläfli symbol	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	{2,6}	{2,7}	{2,8}	{2,9}	{2,10}	{2,11}	{2,12}	...	{2,∞}
Coxeter diagram													...
Faces and edges	1	2	3	4	5	6	7	8	9	10	11	12	...	∞
Vertices	2												...	2
Vertex config.	2	2.2	23	24	25	26	27	28	29	210	211	212	...	2∞

The ${\displaystyle 2n}$ digonal spherical lune faces of a ${\displaystyle 2n}$-hosohedron, ${\displaystyle \{2,2n\}}$, represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry ${\displaystyle C_{nv}}$, ${\displaystyle [n]}$, ${\displaystyle (*nn)}$, order ${\displaystyle 2n}$. The reflection domains can be shown by alternately colored lunes as mirror images.

Bisecting each lune into two spherical triangles creates an ${\displaystyle n}$-gonal bipyramid, which represents the dihedral symmetry ${\displaystyle D_{nh}}$, order ${\displaystyle 4n}$.

Different representations of the kaleidoscopic symmetry of certain small hosohedra

Symmetry (order ${\displaystyle 2n}$)	Schönflies notation	${\displaystyle C_{nv}}$	${\displaystyle C_{1v}}$	${\displaystyle C_{2v}}$	${\displaystyle C_{3v}}$	${\displaystyle C_{4v}}$	${\displaystyle C_{5v}}$	${\displaystyle C_{6v}}$
	Orbifold notation	${\displaystyle (*nn)}$	${\displaystyle (*11)}$	${\displaystyle (*22)}$	${\displaystyle (*33)}$	${\displaystyle (*44)}$	${\displaystyle (*55)}$	${\displaystyle (*66)}$
	Coxeter diagram
		${\displaystyle [n]}$	${\displaystyle [\,\,]}$	${\displaystyle [2]}$	${\displaystyle [3]}$	${\displaystyle [4]}$	${\displaystyle [5]}$	${\displaystyle [6]}$
${\displaystyle 2n}$-gonal hosohedron	Schläfli symbol	${\displaystyle \{2,2n\}}$	${\displaystyle \{2,2\}}$	${\displaystyle \{2,4\}}$	${\displaystyle \{2,6\}}$	${\displaystyle \{2,8\}}$	${\displaystyle \{2,10\}}$	${\displaystyle \{2,12\}}$
	Alternately colored fundamental domains

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.[3]

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.

In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

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.

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”.[4] It was introduced by Vito Caravelli in the eighteenth century.[5]

Wikimedia Commons has media related to Hosohedra.

• Polyhedron
• Polytope

• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0
• Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc., ISBN 0-486-61480-8

• Weisstein, Eric W. "Hosohedron". MathWorld.