 Dihedron

Set of regular ngonal dihedrons
Example hexagonal dihedron on a sphereType Regular polyhedron
or spherical tilingFaces 2 ngons Edges n Vertices n Schläfli symbol {n,2} Vertex configuration n^{2} Coxeter–Dynkin diagram Wythoff symbol 2  n 2 Symmetry group D_{nh}, [2,n], (*22n) Dual polyhedron hosohedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In threedimensional Euclidean space, it is degenerate if its faces are flat, while in threedimensional 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 ngonal dihedron is the ngonal hosohedron, where n digon faces share two vertices.
Contents
As a polyhedron
A dihedron can be considered a degenerate prism consisting of two (planar) nsided polygons connected "backtoback", 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 nprism.
As a tiling on a sphere
As a spherical tiling, a dihedron can exist as nondegenerate form, with two nsided 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 selfdual, and is both a hosohedron and a dihedron.
Regular dihedron examples: (spherical tilings)
Ditopes
A regular ditope is an ndimensional 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
Notes
 ^ Gausmann, Evelise; Roland Lehoucq, JeanPierre Luminet, JeanPhilippe Uzan, Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces". Classical and Quantum Gravity 18: 5155–5186. arXiv:grqc/0106033. doi:10.1088/02649381/18/23/311.
References
 Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0486614808
 Weisstein, Eric W., "Dihedron" from MathWorld.
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(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular dihedron · hosohedronDihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics.Categories:
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