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Dihedron

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Title: Dihedron  
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Subject: Octahedron, Cube, Spherical polyhedron, List of mathematical shapes, Tetrahedron
Collection: Polyhedra
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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).[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.

Contents

  • As a polyhedron 1
  • As a tiling on a sphere 2
  • Apeirogonal dihedron 3
  • Ditopes 4
  • See also 5
  • Notes 6
  • References 7

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.[3]

See also

Notes

  1. ^
  2. ^ Coxeter, Regular polytopes, p. 12
  3. ^ Regular Abstract polytopes, p. 158

References


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