Hosohedron

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

A regular n-gonal hosohedron has Schläfli symbol {2,n}, with each spherical lune having internal angle ⁠2π/n⁠radians (⁠360/n⁠ degrees).

Hosohedra as regular polyhedra

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

N 2 = 4 n 2 m + 2 n − m n . {\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

N 2 = 4 n 2 × 2 + 2 n − 2 n = n , {\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	Henagonal hosohedron	Digonal hosohedron	Trigonal hosohedron	Square hosohedron	Pentagonal hosohedron	...	Apeirogonal hosohedron
Tiling image						...
Schläfli symbol	{2,1}	{2,2}	{2,3}	{2,4}	{2,5}	...	{2,∞}
Coxeter diagram						...
Faces and edges	1	2	3	4	5	...	∞
Vertices	2	2	2	2	2	...	2
Vertex config.	2	2.2	23	24	25	...	2∞

Kaleidoscopic symmetry

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

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

Different representations of the kaleidoscopic symmetry of certain small hosohedra
Symmetry (order 2 n {\displaystyle 2n})	Schönflies notation	C n v {\displaystyle C_{nv}}	C 1 v {\displaystyle C_{1v}}	C 2 v {\displaystyle C_{2v}}	C 3 v {\displaystyle C_{3v}}	C 4 v {\displaystyle C_{4v}}	C 5 v {\displaystyle C_{5v}}	C 6 v {\displaystyle C_{6v}}
Orbifold notation	( ∗ n n ) {\displaystyle (*nn)}	( ∗ 11 ) {\displaystyle (*11)}	( ∗ 22 ) {\displaystyle (*22)}	( ∗ 33 ) {\displaystyle (*33)}	( ∗ 44 ) {\displaystyle (*44)}	( ∗ 55 ) {\displaystyle (*55)}	( ∗ 66 ) {\displaystyle (*66)}
Coxeter diagram
[ n ] {\displaystyle [n]}	[ ] {\displaystyle [\,\,]}	[ 2 ] {\displaystyle [2]}	[ 3 ] {\displaystyle [3]}	[ 4 ] {\displaystyle [4]}	[ 5 ] {\displaystyle [5]}	[ 6 ] {\displaystyle [6]}
2 n {\displaystyle 2n}-gonal hosohedron	Schläfli symbol	{ 2 , 2 n } {\displaystyle \{2,2n\}}	{ 2 , 2 } {\displaystyle \{2,2\}}	{ 2 , 4 } {\displaystyle \{2,4\}}	{ 2 , 6 } {\displaystyle \{2,6\}}	{ 2 , 8 } {\displaystyle \{2,8\}}	{ 2 , 10 } {\displaystyle \{2,10\}}	{ 2 , 12 } {\displaystyle \{2,12\}}
Alternately colored fundamental domains

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:

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.

External links

Weisstein, Eric W. . MathWorld.