| Set of regularn-gonal dihedra | |
|---|---|
 Example hexagonal dihedron on a sphere | |
| Type | regularpolyhedron orspherical tiling |
| Faces | 2n-gons |
| Edges | n |
| Vertices | n |
| Vertex configuration | n.n |
| Wythoff symbol | 2 |n 2 |
| 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 |
Adihedron is a type ofpolyhedron, made of twopolygon faces which share the same set ofnedges. In three-dimensionalEuclidean space, it isdegenerate if 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 space L(p,q).[1] Dihedra have also been calledbihedra,[2]flat polyhedra,[3] ordoubly covered polygons.[3]
As aspherical tiling, adihedron can exist as nondegenerate form, with twon-sided faces covering the sphere, each face being ahemisphere, and vertices on agreat circle. It isregular if the vertices are equally spaced.
Thedual of ann-gonal dihedron is ann-gonalhosohedron, wherendigon faces share two vertices.
Adihedron can be considered a degenerateprism whose two (planar)n-sidedpolygon 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 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 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 as convex polyhedra.[4]
Some dihedra can arise as lower limit members of other polyhedra families: aprism withdigon bases would be a square dihedron, and apyramid with a digon base would be a triangular dihedron.
Aregular dihedron, with Schläfli symbol {n,2}, is made of tworegular polygons, each withSchläfli symbol {n}.[5]
Aspherical dihedron is made of twospherical polygons which share the same set ofn vertices, on agreat circle equator; each polygon of a spherical dihedron fills ahemisphere.
Aregular spherical dihedron is made of two regular spherical polygons which share the same set ofn vertices, equally spaced on agreat circle equator.
The regular polyhedron {2,2} is self-dual, and is both ahosohedron and a dihedron.
| 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 | ... | ∞.∞ |
Asn tends to infinity, ann-gonal dihedron becomes anapeirogonal dihedron as a 2-dimensional tessellation:
A regularditope is 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.[6]