Dihedron

A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In three-dimensional Euclidean space, 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 L(p,q).

Usually a regular dihedron is implied (two regular polygons) and this gives it a Schläfli symbol as {n,2}. Each polygon fills a hemisphere, with a regular n-gon on a great circle equator between them.

The dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices.

As a polyhedron

A dihedron can be considered a degenerate prism consisting of two (planar) n-sided polygons connected "back-to-back", so that the resulting object has no depth.

As a tiling on a sphere

As a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided 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 self-dual, and is both a hosohedron and a dihedron.

Regular dihedra: (spherical tilings)

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 2-dimensional tessellation:

Ditopes

A regular ditope is an n-dimensional 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.

See also

• Polyhedron
• Polytope

References

• Coxeter, H.S.M.; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
• Weisstein, Eric W., "Dihedron", MathWorld.