Stratonovich integral: Difference between revisions

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{{Infobox polyhedron
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| name          =Set of regular ''n''-gonal hosohedra
| image        =Hexagonal hosohedron.png
| caption      =Example hexagonal hosohedron on a sphere
| type          =[[Regular polyhedron]] or [[spherical tiling]]
| euler        = 2
| faces        =''n'' [[digon]]s
| edges        =''n''
| vertices      =2
| vertex_config =''2''<sup>n</sup>
| schläfli      ={2,''n''}
| wythoff      =''n'' {{!}} 2 2
| coxeter      ={{CDD|node|n|node|2|node_1}}
| symmetry      =D<sub>''n''h</sub>, [2,n], (*22n), order 4n
| rotsymmetry  =D<sub>''n''</sub>, [2,n]<sup>+</sup>, (22n), order 2n
| surface_area  =
| volume        =
| angle        =
| dual          =[[dihedron]]
| properties    =
| vertex_figure =  
| net          =}}
[[Image:BeachBall.jpg|thumb|This [[beach ball]] shows a hosohedron with six lune faces, if the white circles on the ends are removed.]]
In [[geometry]], an [[Polygon|''n''-gonal]] '''hosohedron''' is a tessellation of [[Lune (mathematics)|lunes]] on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has [[Schläfli symbol]] {2,&nbsp;''n''}.
 
== Hosohedra as regular polyhedra ==
For a regular polyhedron whose Schläfli symbol is {''m'',&nbsp;''n''}, the number of polygonal faces may be found by:
 
:<math>N_2=\frac{4n}{2m+2n-mn}</math>
 
The [[Platonic solid]]s 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 [[digon]]s (2-gons) can be represented as spherical lunes, having non-zero [[area (geometry)|area]]. Allowing ''m'' = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2,&nbsp;''n''} is represented as ''n'' abutting lunes, with interior angles of 2π/''n''. All these lunes share two common vertices.
 
{| class="wikitable" width="320"
|[[File:Trigonal_hosohedron.png|160px]]<br />A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.
|[[Image:4hosohedron.svg|160px]]<br />A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.
|}
 
{|class="wikitable"
|+ Family of regular hosohedra
|-
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!...
|-
!{{CDD|node_1|2|node}}<BR>{2,1}
!{{CDD|node_1|2|node|2|node}}<BR>{2,2}
!{{CDD|node_1|2|node|3|node}}<BR>{2,3}
!{{CDD|node_1|2|node|4|node}}<BR>{2,4}
!{{CDD|node_1|2|node|5|node}}<BR>{2,5}
!{{CDD|node_1|2|node|6|node}}<BR>{2,6}
!{{CDD|node_1|2|node|7|node}}<BR>{2,7}
!{{CDD|node_1|2|node|8|node}}<BR>{2,8}
!{{CDD|node_1|2|node|9|node}}<BR>{2,9}
!{{CDD|node_1|2|node|1x|0x|node}}<BR>{2,10}
!{{CDD|node_1|2|node|1x|1x|node}}<BR>{2,11}
!{{CDD|node_1|2|node|1x|2x|node}}<BR>{2,12}
|-
|[[File:Spherical henagonal hosohedron.png|50px]]
|[[File:Spherical digonal hosohedron.png|50px]]
|[[Image:Spherical trigonal hosohedron.png|50px]]
|[[Image:Spherical square hosohedron.png|50px]]
|[[Image:Spherical pentagonal hosohedron.png|50px]]
|[[Image:Spherical hexagonal hosohedron.png|50px]]
|[[Image:Spherical heptagonal hosohedron.png|50px]]
|[[Image:Spherical octagonal hosohedron.png|50px]]
|[[Image:Spherical enneagonal hosohedron.png|50px]]
|[[Image:Spherical decagonal hosohedron.png|50px]]
|[[Image:Spherical hendecagonal hosohedron.png|50px]]
|[[Image:Spherical dodecagonal hosohedron.png|50px]]
|}
 
== Kalidescopic symmetry ==
The digonal faces of a 2''n''-hosohedron, {2,2n}, represents the fundamental domains of [[dihedral symmetry in three dimensions]]: C<sub>nv</sub>, [n], (*nn), order 2''n''. The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates [[Bipyramid#Symmetry|bipyramids]] and define [[dihedral symmetry]] D<sub>nh</sub>, order 4''n''.
 
{|class="wikitable" width=480
!Symmetry
!C<sub>1v</sub>
!C<sub>2v</sub>
!C<sub>3v</sub>
!C<sub>4v</sub>
!C<sub>5v</sub>
!C<sub>6v</sub>
|-
!Hosohedron
!{2,2}
!{2,4}
!{2,6}
!{2,8}
!{2,10}
!{2,12}
|-
!Fundamental domains
|[[Image:Spherical digonal hosohedron2.png|80px]]
|[[Image:Spherical square hosohedron2.png|80px]]
|[[Image:Spherical hexagonal hosohedron2.png|80px]]
|[[Image:Spherical octagonal hosohedron2.png|80px]]
|[[Image:Spherical decagonal hosohedron2.png|80px]]
|[[Image:Spherical dodecagonal hosohedron2.png|80px]]
|}
 
== Relationship with the Steinmetz solid ==
The tetragonal hosohedron is topologically equivalent to the [[Steinmetz solid#Bicylinder|bicylinder Steinmetz solid]], the intersection of two cylinders at right-angles.<ref>{{mathworld|urlname=SteinmetzSolid|title=Steinmetz Solid}}</ref>
 
== Derivative polyhedra ==
The [[dual polyhedron|dual]] of the n-gonal hosohedron {2,&nbsp;''n''} is the ''n''-gonal [[dihedron]], {''n'',&nbsp;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 polyhedron|truncated]] variation. The truncated ''n''-gonal hosohedron is the n-gonal [[prism (geometry)|prism]]. 
 
== Hosotopes ==
[[dimension|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” was coined by [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]], and possibly derives from the Greek ὅσος (''osos/hosos'') “as many”, the idea being that a hosohedron can have “'''as many''' faces as desired”. <ref name="Schwartzman1994">{{cite book|author=Steven Schwartzman|title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English|url=http://books.google.com/books?id=SRw4PevE4zUC&pg=PA109|date=1 January 1994|publisher=MAA|isbn=978-0-88385-511-9|pages=108–109}}</ref>
 
== See also ==
{{Commonscat|Hosohedra}}
*[[Polyhedron]]
*[[Polytope]]
 
== References ==
{{reflist}}
*Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
*{{mathworld | urlname = Hosohedron | title = Hosohedron}}
 
{{Polyhedron navigator}}
{{polyhedra}}
 
[[Category:Polyhedra]]
[[Category:Tessellation]]

Latest revision as of 19:33, 11 August 2014

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