View source for Dihedron

{{Short description|Polyhedron with 2 faces}}
{{Infobox polyhedron
| name          =Set of regular ''n''-gonal dihedra
| image         =Hexagonal dihedron.svg
| caption       =Example hexagonal dihedron on a sphere
| type          =regular [[polyhedron]] or [[spherical tiling]]
| euler         =
| faces         =2 ''n''-gons
| edges         =''n''
| vertices      =''n''
| vertex_config =''n''.''n''
| schläfli      ={''n'',2}
| wythoff       =2 {{!}} ''n'' 2
| coxeter       ={{CDD|node_1|n|node|2x|node}}
| symmetry      =D<sub>''n''h</sub>, [2,''n''], (*22''n''), order 4''n''
| rotsymmetry   =D<sub>''n''</sub>, [2,''n'']<sup>+</sup>, (22''n''), order 2''n''
| surface_area  =
| volume        =
| angle         =
| dual          =[[Regular polyhedron|regular]] ''n''-gonal [[hosohedron]]
| properties    =
| vertex_figure =
| net           =}}
A '''dihedron''' (pl. '''dihedra''') is a type of [[polyhedron]], made of two [[polygon]] faces which share the same set of ''n'' [[Polyhedron|edges]]. In three-dimensional [[Euclidean space]], it is [[Mathematical degeneracy|degenerate]] if its faces are flat, while in three-dimensional [[Spherical 3-manifold#Cyclic case .28lens spaces.29|spherical space]], a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a [[lens space]] L(''p'',''q'').<ref name="Gausmann2001">{{cite journal |last=Gausmann |first=Evelise |author2=Roland Lehoucq |author3=Jean-Pierre Luminet |author4=Jean-Philippe Uzan |author5=Jeffrey Weeks |title=Topological Lensing in Spherical Spaces |journal=Classical and Quantum Gravity |volume=18 |issue= 23|pages=5155–5186 |year=2001 |arxiv=gr-qc/0106033 |doi=10.1088/0264-9381/18/23/311|bibcode=2001CQGra..18.5155G |s2cid=34259877 }}</ref> Dihedra have also been called '''bihedra''',<ref>{{citation
 | last = Kántor
 | first = S.
 | issue = 1
 | journal = Beiträge zur Algebra und Geometrie
 | mr = 1990989
 | pages = 145–154
 | title = On the volume of unbounded polyhedra in the hyperbolic space
 | url = http://www.emis.de/journals/BAG/vol.44/no.1/b44h1kan.pdf
 | volume = 44
 | year = 2003
 | access-date = 2017-02-14
 | archive-date = 2017-02-15
 | archive-url = https://web.archive.org/web/20170215123527/http://www.emis.de/journals/BAG/vol.44/no.1/b44h1kan.pdf
 | url-status = dead
 }}.</ref> '''flat polyhedra''',<ref name="zip">{{citation|arxiv=1010.2450|title= Flat zipper-unfolding pairs for Platonic solids |first=Joseph|last=O'Rourke|year=2010|author-link=Joseph O'Rourke (professor)|bibcode=2010arXiv1010.2450O}}</ref> or '''doubly covered polygons'''.<ref name="zip"/>

As a [[spherical tiling]], a '''dihedron''' can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a [[Sphere|hemisphere]], and vertices on a [[great circle]]. It is '''regular''' if the vertices are equally spaced.

The [[Dual polyhedron|dual]] of an ''n''-gonal dihedron is an ''n''-gonal [[hosohedron]], where ''n'' [[digon]] faces share two vertices.

==As a flat-faced polyhedron==
A '''dihedron''' can be considered a degenerate [[Prism (geometry)|prism]] whose two (planar) ''n''-sided [[polygon]] bases are connected "back-to-back", so that the resulting object has no depth. The polygons must be congruent, but glued in such a way that one is the mirror image of the other. This applies only if the distance between the two faces is zero; for a distance larger than zero, the faces are infinite polygons (a bit like the [[apeirogonal hosohedron]]'s digon faces, having a width larger than zero, are infinite stripes).

Dihedra can arise from [[Alexandrov's uniqueness theorem]], which characterizes the distances on the surface of any convex polyhedron as being locally Euclidean except at a finite number of points with positive [[angular defect]] summing to 4{{pi}}. This characterization holds also for the distances on the surface of a dihedron, so the statement of Alexandrov's theorem requires that dihedra be considered as convex polyhedra.<ref>{{citation|arxiv=1007.2016|title=On flat polyhedra deriving from Alexandrov's theorem|first=Joseph|last=O'Rourke|year=2010|author-link=Joseph O'Rourke (professor)|bibcode=2010arXiv1007.2016O}}</ref>

Some dihedra can arise as lower limit members of other polyhedra families: a [[Prism (geometry)|prism]] with [[digon]] bases would be a square dihedron, and a [[Pyramid (geometry)|pyramid]] with a digon base would be a triangular dihedron.

A '''regular dihedron''', with Schläfli symbol {''n'',2}, is made of two [[regular polygon]]s, each with [[Schläfli symbol]] {''n''}.<ref>{{citation|last=Coxeter|first=H. S. M.|title=Regular Polytopes|date=January 1973|url=https://archive.org/details/regularpolytopes0000coxe/page/12|page=[https://archive.org/details/regularpolytopes0000coxe/page/12 12]|edition=3rd|publisher=Dover Publications Inc.|isbn=0-486-61480-8|author-link=Harold Scott MacDonald Coxeter}}</ref>

==As a tiling of the sphere==
A '''spherical dihedron''' is made of two [[Spherical trigonometry|spherical polygons]] which share the same set of ''n'' vertices, on a [[great circle]] equator; each polygon of a spherical dihedron fills a [[Sphere#Hemisphere|hemisphere]].

A '''regular spherical dihedron''' is made of two regular spherical polygons which share the same set of ''n'' vertices, equally spaced on a [[great circle]] equator.

The regular polyhedron {2,2} is self-dual, and is both a [[hosohedron]] and a dihedron.

{{Regular dihedral tilings}}

==Apeirogonal dihedron==
As ''n'' tends to infinity, an ''n''-gonal dihedron becomes an [[apeirogonal dihedron]] as a 2-dimensional tessellation:<br>[[File:Apeirogonal tiling.svg|frameless]]

==Ditopes==
A regular ''ditope'' is an ''n''-dimensional analogue of a dihedron, with Schläfli symbol {''p'',...,''q'',''r'',2}. It has two [[Facet (geometry)|facets]], {''p'',...,''q'',''r''}, which share all [[Ridge (geometry)|ridges]], {''p'',...,''q''} in common.<ref>{{citation | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | title = Abstract Regular Polytopes | edition = 1st | publisher = [[Cambridge University Press]] | isbn = 0-521-81496-0 | date = December 2002 | page = [https://archive.org/details/abstractregularp0000mcmu/page/158 158] | url = https://archive.org/details/abstractregularp0000mcmu/page/158 }}</ref>

==See also==
*[[Dihedral group]]
*[[Polyhedron]]
*[[Polytope]]

==References==
{{reflist|30em}}

==External links==
*{{mathworld |urlname=Dihedron |title=Dihedron}}

{{Polyhedra}}
{{Polyhedron navigator}}

[[Category:Polyhedra]]
[[Category:Regular polyhedra]]