A dihedron (pl. dihedra) is a type of

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

, made of two

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

faces which share the same set of ''n'' edges. In three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, it is degenerate if its faces are flat, while in three-dimensional 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 A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized ...

L(''p'',''q''). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons. As a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called ''spherical polygons''. A polyhedron whose vertices are equi ...

, a dihedron can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a

hemisphere Hemisphere may refer to: In geometry * Hemisphere (geometry), a half of a sphere As half of Earth or any spherical astronomical object * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemi ...

, and vertices on a

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...

. It is regular if the vertices are equally spaced. The dual of an ''n''-gonal dihedron is an ''n''-gonal

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

, where ''n''

digon In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

faces share two vertices.

As a flat-faced polyhedron

A dihedron can be considered a degenerate

prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

whose two (planar) ''n''-sided

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 Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each othe ...

, 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 In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the ''exces ...

summing to 4. 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: a

with

bases would be a square dihedron, and a

pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

with a digon base would be a triangular dihedron. A regular dihedron, with Schläfli symbol , is made of two

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s, each with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

As a tiling of the sphere

A spherical dihedron is made of two spherical polygons which share the same set of ''n'' vertices, on a

equator; each polygon of a spherical dihedron fills a

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

equator. The regular polyhedron is self-dual, and is both a

and a dihedron.

Apeirogonal dihedron

As ''n'' tends to infinity, an ''n''-gonal dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation:
Apeirogonal tiling

Ditopes

A regular ''ditope'' is an ''n''-dimensional analogue of a dihedron, with Schläfli symbol . It has two facets, , which share all ridges, in common.

References

External links