Set of regular n-gonal dihedra | |
---|---|

Type | regular polyhedron or spherical tiling |

Faces | 2 n-gons |

Edges | n |

Vertices | n |

Vertex configuration | n.n |

Wythoff symbol | 2 | n 2 |

Schläfli symbol | {n,2} |

Coxeter diagram | |

Symmetry group | D_{nh}, [2,n], (*22n), order 4n |

Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |

Dual polyhedron | regular n-gonal hosohedron |

A **dihedron** is a type of polyhedron, made of two polygon faces which share the same set of *n* 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] } Dihedra have also been called **bihedra**,^{ [2] }**flat polyhedra**,^{ [3] } or **doubly covered polygons**.^{ [3] }

- As a flat-faced polyhedron
- As a tiling of the sphere
- Apeirogonal dihedron
- Ditopes
- See also
- References
- External links

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 on a great circle. It is **regular** if the vertices are equally spaced.

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

A **dihedron** can be considered a degenerate prism whose two (planar) *n*-sided polygon bases are connected "back-to-back", so that the resulting object has no depth. The polygons must be congruent, but glued in such a way that one is the mirror image of the other. This applies only if the distance between the two faces is zero; for a distance larger than zero, the faces are infinite polygons (a bit like the apeirogonal hosohedron's digon faces, having a width larger than zero, are infinite stripes).

Dihedra can arise from Alexandrov's uniqueness theorem, which characterizes the distances on the surface of any convex polyhedron as being locally Euclidean except at a finite number of points with positive angular defect summing to 4π. This characterization holds also for the distances on the surface of a dihedron, so the statement of Alexandrov's theorem requires that dihedra be considered as convex polyhedra.^{ [4] }

Some dihedra can arise as lower limit members of other polyhedra families: a prism with digon bases would be a square dihedron, and a pyramid with a digon base would be a triangular dihedron.

A **regular dihedron**, with Schläfli symbol {*n*,2}, is made of two regular polygons, each with Schläfli symbol {*n*}.^{ [5] }

A **spherical dihedron** is made of two spherical polygons which share the same set of *n* vertices, on a great circle equator; each polygon of a spherical dihedron fills a hemisphere.

A **regular spherical dihedron** is made of two regular spherical polygons which share the same set of *n* vertices, equally spaced on a great circle equator.

The regular polyhedron {2,2} is self-dual, and is both a hosohedron and a dihedron.

Space | Spherical | Euclidean | ||||||
---|---|---|---|---|---|---|---|---|

Tiling name | (Hengonal) Monogonal dihedron | Digonal dihedron | (Triangular) Trigonal dihedron | (Tetragonal) Square dihedron | Pentagonal dihedron | Hexagonal dihedron | ... | Apeirogonal dihedron |

Tiling image | ... | |||||||

Schläfli symbol | {1,2} | {2,2} | {3,2} | {4,2} | {5,2} | {6,2} | ... | {∞,2} |

Coxeter diagram | ... | |||||||

Faces | 2 {1} | 2 {2} | 2 {3} | 2 {4} | 2 {5} | 2 {6} | ... | 2 {∞} |

Edges and vertices | 1 | 2 | 3 | 4 | 5 | 6 | ... | ∞ |

Vertex config. | 1.1 | 2.2 | 3.3 | 4.4 | 5.5 | 6.6 | ... | ∞.∞ |

As *n* tends to infinity, an *n*-gonal dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation:

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.^{ [6] }

In geometry, every polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or **n-polytope**. In this context, "flat sides" means that the sides of a (*k* + 1)-polytope consist of k-polytopes that may have (*k* – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a **prism** is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

In geometry, a **net** of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.

In spherical geometry, an n-gonal**hosohedron** is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

In geometry, a **uniform polyhedron** has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, the **hexagonal prism** is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices.

In geometry, a **digon** is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space.

In geometry, a **monogon**, also known as a **henagon**, is a polygon with one edge and one vertex. It has Schläfli symbol {1}.

In geometry, a **star polyhedron** is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.

In geometry, a **spherical polyhedron** or **spherical tiling** is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.

In geometry, a **toroidal polyhedron** is a polyhedron which is also a toroid, having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

In geometry, a (globally) **projective polyhedron** is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.

The **Alexandrov uniqueness theorem** is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.

- ↑ Gausmann, Evelise; Roland Lehoucq; Jean-Pierre Luminet; Jean-Philippe Uzan; Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces".
*Classical and Quantum Gravity*.**18**(23): 5155–5186. arXiv: gr-qc/0106033 . Bibcode:2001CQGra..18.5155G. doi:10.1088/0264-9381/18/23/311. S2CID 34259877. - ↑ Kántor, S. (2003), "On the volume of unbounded polyhedra in the hyperbolic space" (PDF),
*Beiträge zur Algebra und Geometrie*,**44**(1): 145–154, MR 1990989, archived from the original (PDF) on 2017-02-15, retrieved 2017-02-14. - 1 2 O'Rourke, Joseph (2010),
*Flat zipper-unfolding pairs for Platonic solids*, arXiv: 1010.2450 , Bibcode:2010arXiv1010.2450O - ↑ O'Rourke, Joseph (2010),
*On flat polyhedra deriving from Alexandrov's theorem*, arXiv: 1007.2016 , Bibcode:2010arXiv1007.2016O - ↑ Coxeter, H. S. M. (January 1973),
*Regular Polytopes*(3rd ed.), Dover Publications Inc., p. 12, ISBN 0-486-61480-8 - ↑ McMullen, Peter; Schulte, Egon (December 2002),
*Abstract Regular Polytopes*(1st ed.), Cambridge University Press, p. 158, ISBN 0-521-81496-0

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