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Set of regular n-gonal dihedra
Hexagonal dihedron.png
Example hexagonal dihedron on a sphere
Type Regular polyhedron or spherical tiling
Faces 2 n-gons
Edges n
Vertices n
Vertex configuration n.n
Schläfli symbol {n,2}
Wythoff symbol 2 | n 2
Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 2x.pngCDel node.png
Symmetry group Dnh, [2,n], (*22n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
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}. Each polygon fills a hemisphere, with a regular n-gon on a great circle equator between them.[2]

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

As a polyhedron[edit]

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[edit]

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 Digonal dihedron.png Trigonal dihedron.png Tetragonal dihedron.png Pentagonal dihedron.png Hexagonal dihedron.png
Schläfli {2,2} {3,2} {4,2} {5,2} {6,2}...
Coxeter CDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png
Faces 2 {2} 2 {3} 2 {4} 2 {5} 2 {6}
Edges and
2 3 4 5 6

Apeirogonal dihedron[edit]

In the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation:

Apeirogonal tiling.png


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.[3]

See also[edit]


  1. ^ Gausmann, Evelise; Roland Lehoucq; Jean-Pierre Luminet; Jean-Philippe Uzan; Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces". Classical and Quantum Gravity 18: 5155–5186. arXiv:gr-qc/0106033. doi:10.1088/0264-9381/18/23/311. 
  2. ^ Coxeter, Regular polytopes, p. 12
  3. ^ Regular Abstract polytopes, p. 158