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.
Set of regular n-gonal hosohedra | |
---|---|
Type | regular polyhedron or spherical tiling |
Faces | n digons |
Edges | n |
Vertices | 2 |
Euler char. | 2 |
Vertex configuration | 2^{n} |
Wythoff symbol | n | 2 2 |
Schläfli symbol | {2,n} |
Coxeter diagram | |
Symmetry group | D_{nh} [2,n] (*22n) order 4n |
Rotation group | D_{n} [2,n]^{+} (22n) order 2n |
Dual polyhedron | regular n-gonal dihedron |
A regular n-gonal hosohedron has Schläfli symbol {2,n}, with each spherical lune having internal angle 2π/nradians (360/n degrees).^{[1]}^{[2]}
For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :
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
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. |
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 | 2^{3} | 2^{4} | 2^{5} | ... | 2^{∞} |
The digonal spherical lune faces of a -hosohedron, , represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry , , , order . The reflection domains can be shown by alternately colored lunes as mirror images.
Bisecting each lune into two spherical triangles creates an -gonal bipyramid, which represents the dihedral symmetry , order .
Symmetry (order ) | Schönflies notation | |||||||
---|---|---|---|---|---|---|---|---|
Orbifold notation | ||||||||
Coxeter diagram | ||||||||
-gonal hosohedron | Schläfli symbol | |||||||
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]}
The hosohedron {2,p} (in a slightly distorted form) was named by Vito Caravelli (1724–1800) …