Dihedron

Adihedronis a type ofpolyhedron,made of twopolygonfaces which share the same set ofnedges.In three-dimensionalEuclidean space,it isdegenerateif its faces are flat, while in three-dimensionalspherical space,a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of alens spaceL(p,q).Dihedra have also been calledbihedra,flat polyhedra,ordoubly covered polygons.

Set of regularn-gonal dihedra
Example hexagonal dihedron on a sphere
Type	regularpolyhedronorspherical tiling
Faces	2n-gons
Edges	n
Vertices	n
Vertex configuration	n.n
Wythoff symbol	2 |n2
Schläfli symbol	{n,2}
Coxeter diagram
Symmetry group	Dnh,[2,n], (*22n), order 4n
Rotation group	Dn,[2,n]+,(22n), order 2n
Dual polyhedron	regularn-gonalhosohedron

As aspherical tiling,adihedroncan exist as nondegenerate form, with twon-sided faces covering the sphere, each face being ahemisphere,and vertices on agreat circle.It isregularif the vertices are equally spaced.

Thedualof ann-gonal dihedron is ann-gonalhosohedron,wherendigonfaces share two vertices.

As a flat-faced polyhedron

Adihedroncan be considered a degenerateprismwhose two (planar)n-sidedpolygonbases 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 theapeirogonal hosohedron's digon faces, having a width larger than zero, are infinite stripes).

Dihedra can arise fromAlexandrov'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 positiveangular defectsumming 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 as convex polyhedra.

Some dihedra can arise as lower limit members of other polyhedra families: aprismwithdigonbases would be a square dihedron, and apyramidwith a digon base would be a triangular dihedron.

Aregular dihedron,with Schläfli symbol {n,2}, is made of tworegular polygons,each withSchläfli symbol{n}.

As a tiling of the sphere

Aspherical dihedronis made of twospherical polygonswhich share the same set ofnvertices, on agreat circleequator; each polygon of a spherical dihedron fills ahemisphere.

Aregular spherical dihedronis made of two regular spherical polygons which share the same set ofnvertices, equally spaced on agreat circleequator.

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

Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings:nn

Space	Spherical						Euclidean
Tiling name	Monogonal dihedron	Digonal dihedron	Trigonal dihedron	Square dihedron	Pentagonal dihedron	...	Apeirogonal dihedron
Tiling image						...
Schläfli symbol	{1,2}	{2,2}	{3,2}	{4,2}	{5,2}	...	{∞,2}
Coxeter diagram						...
Faces	2{1}	2{2}	2{3}	2{4}	2{5}	...	2{∞}
Edges and vertices	1	2	3	4	5	...	∞
Vertex config.	1.1	2.2	3.3	4.4	5.5	...	∞.∞

Apeirogonal dihedron

Asntends to infinity, ann-gonal dihedron becomes anapeirogonal dihedronas a 2-dimensional tessellation:

Ditopes

A regularditopeis ann-dimensional analogue of a dihedron, with Schläfli symbol {p,...,q,r,2}. It has twofacets,{p,...,q,r}, which share allridges,{p,...,q} in common.

See also

• Dihedral group
• Polyhedron
• Polytope

External links

• Weisstein, Eric W."Dihedron".MathWorld.