Dihedron

Set of regular p-gonal dihedrons
Example hexagonal dihedron on a sphere
Type	Regular polyhedron or spherical tiling
Faces	2 p-gons
Edges	p
Vertices	p
Schläfli symbol	{p,2}
Vertex configuration	p2
Coxeter–Dynkin diagram
Wythoff symbol	2 | p 2
Symmetry group	Dihedral (Dph)
Dual polyhedron	hosohedron

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).^[1]

Usually a regular dihedron is implied (two regular polygons) and this gives it a Schläfli symbol as {n, 2}.

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.

From a Wythoff construction on dihedral symmetry, a truncation operation on a regular {n,2} dihedron transforms it into a 4.4.n n-prism.

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 dihedron examples: (spherical tilings)

Ditopes

A ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol {p,2,...,2}. It has two facets which share all ridges 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" from MathWorld.

1. ^ Gausmann, Evelise; Roland Lehoucq, Jean-Pierre Luminet, Jean-Philippe Uzan, Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces". Classical and Quantum Gravity (Insitute of Physics Publishing) 18: 5155-5186. http://arxiv.org/abs/gr-qc/0106033v1. Retrieved 2009-11-02.