From Wikipedia, the free encyclopedia
Spherical polyhedron composed of lunes
This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.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).[1][2]
Hosohedra as regular polyhedra[edit]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.
Kaleidoscopic symmetry[edit]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[edit]The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.[3]
Derivative polyhedra[edit]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[edit]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.
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4