Dihedron
Set of regular ngonal dihedra  

Example hexagonal dihedron on a sphere  
Type  Regular polyhedron or spherical tiling 
Faces  2 ngons 
Edges  n 
Vertices  n 
Vertex configuration  n.n 
Wythoff symbol  2  n 2 
Schläfli symbol  {n,2} 
Coxeter diagram 

Symmetry group  D_{nh}, [2,n], (*22n), order 4n 
Rotation group  D_{n}, [2,n]^{+}, (22n), order 2n 
Dual polyhedron  hosohedron 
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In threedimensional Euclidean space, it is degenerate if its faces are flat, while in threedimensional 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).^{[1]} Dihedra have also been called bihedra,^{[2]} flat polyhedra,^{[3]} or doubly covered polygons.^{[3]}
A regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol {n,2}.^{[4]} As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, with a regular ngon on a great circle equator between them.
The dual of a ngonal dihedron is the ngonal hosohedron, where n digon faces share two vertices.
As a polyhedron
A dihedron can be considered a degenerate prism consisting of two (planar) nsided polygons connected "backtoback", 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.
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 4π. 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 to be convex polyhedra.^{[5]}
As a tiling on a sphere
As a spherical tiling, a dihedron can exist as nondegenerate form, with two nsided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. (It is regular if the vertices are equally spaced.)
The regular polyhedron {2,2} is selfdual, and is both a hosohedron and a dihedron.
Image  
Schläfli  {2,2}  {3,2}  {4,2}  {5,2}  {6,2}... 

Coxeter  
Faces  2 {2}  2 {3}  2 {4}  2 {5}  2 {6} 
Edges and vertices 
2  3  4  5  6 
Apeirogonal dihedron
In the limit the dihedron becomes an apeirogonal dihedron as a 2dimensional tessellation:
Ditopes
A regular ditope is an ndimensional analogue of a dihedron, with Schläfli symbol {p,...q,r,2}. It has two facets, {p,...q,r}, which share all ridges, {p,...q} in common.^{[6]}
See also
References
 ↑ Gausmann, Evelise; Roland Lehoucq; JeanPierre Luminet; JeanPhilippe Uzan; Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces". Classical and Quantum Gravity. 18: 5155–5186. arXiv:grqc/0106033. Bibcode:2001CQGra..18.5155G. doi:10.1088/02649381/18/23/311.
 ↑ Kántor, S. (2003), "On the volume of unbounded polyhedra in the hyperbolic space" (PDF), Beiträge zur Algebra und Geometrie, 44 (1): 145–154, MR 1990989 .
 1 2 O'Rourke, Joseph (2010), Flat zipperunfolding pairs for Platonic solids, arXiv:1010.2450, Bibcode:2010arXiv1010.2450O
 ↑ Coxeter, H. S. M., Regular Polytopes (3rd ed.), Dover Publications Inc., p. 12, ISBN 0486614808
 ↑ O'Rourke, Joseph (2010), On flat polyhedra deriving from Alexandrov's theorem, arXiv:1007.2016, Bibcode:2010arXiv1007.2016O
 ↑ McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, p. 158, ISBN 0521814960