Dihedron
From Wikipedia, the free encyclopedia
| Set of regular p-gonal dihedrons | |
|---|---|
 Example hexagonal dihedron on a sphere |
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| 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. It is degenerate if its faces are flat.
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.
Contents |
[edit] 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.
[edit] As a tiling on a sphere
As a tiling on a sphere, 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)
 {2,2} |
 {3,2} |
 {6,2} |
[edit] 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.
[edit] See also
[edit] References
- Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8
- Eric W. Weisstein, Dihedron at MathWorld.
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