| Set of regular n-gonal dihedrons | |
|---|---|
Example hexagonal dihedron on a sphere |
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| Type | Regular polyhedron or spherical tiling |
| Faces | 2 n-gons |
| Edges | n |
| Vertices | n |
| Schläfli symbol | {n,2} |
| Vertex configuration | n2 |
| Coxeter–Dynkin diagram | |
| Wythoff symbol | 2 | n 2 |
| Symmetry group | Dnh, [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 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.
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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 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)
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.
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