Dihedron

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Set of regular n-gonal dihedra
Hexagonal dihedron.svg
Example hexagonal dihedron on a sphere
Typeregular 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 CDel node 1.pngCDel n.pngCDel node.pngCDel 2x.pngCDel node.png
Symmetry group Dnh, [2,n], (*22n), order 4n
Rotation group Dn, [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]

Contents

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.

As a flat-faced polyhedron

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]

As a tiling of the sphere

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.

Family of regular dihedra · *n22 symmetry mutations of regular dihedral tilings: nn
SpaceSphericalEuclidean
Tiling name(Hengonal)
Monogonal dihedron
Digonal dihedron (Triangular)
Trigonal dihedron
(Tetragonal)
Square dihedron
Pentagonal dihedron Hexagonal dihedron ... Apeirogonal dihedron
Tiling image Monogonal dihedron.svg Digonal dihedron.svg Trigonal dihedron.svg Tetragonal dihedron.svg Pentagonal dihedron.svg Hexagonal dihedron.svg ... Apeirogonal tiling.svg
Schläfli symbol {1,2}{2,2}{3,2}{4,2}{5,2}{6,2}...{∞,2}
Coxeter diagram CDel node 1.pngCDel 1x.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 2x.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 2x.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel node.png...CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png
Faces2 {1} 2 {2} 2 {3} 2 {4} 2 {5} 2 {6} ...2 {∞}
Edges and vertices123456...
Vertex config. 1.12.23.34.45.56.6...∞.∞

Apeirogonal dihedron

As n tends to infinity, an n-gonal dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation:
Apeirogonal tiling.svg

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. [6]

See also

Related Research Articles

Dual polyhedron Polyhedron associated with another by swapping vertices for faces

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.

Polyhedron 3D shape with flat faces, straight edges and sharp corners

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.

Prism (geometry) Solid with parallel bases connected by parallelograms

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.

Schläfli symbol Notation that defines regular polytopes and tessellations

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

Regular polytope Polytope with highest degree of symmetry

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.

Vertex figure Shape made by slicing off a corner of a polytope

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

Net (polyhedron) Edge-joined polygons which fold into a polyhedron

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.

Hosohedron Spherical polyhedron composed of lunes

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

Uniform polyhedron Isogonal polyhedron with regular faces

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

Hexagonal prism Prism with a 6-sided base

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

Digon Polygon with 2 sides and 2 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.

Monogon Polygon with one edge and one vertex

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.

Spherical polyhedron Partition of a spheres surface into polygons

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.

Toroidal polyhedron Partition of a toroidal surface into polygons

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.

References

  1. 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.
  2. 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.
  3. 1 2 O'Rourke, Joseph (2010), Flat zipper-unfolding pairs for Platonic solids, arXiv: 1010.2450 , Bibcode:2010arXiv1010.2450O
  4. O'Rourke, Joseph (2010), On flat polyhedra deriving from Alexandrov's theorem, arXiv: 1007.2016 , Bibcode:2010arXiv1007.2016O
  5. Coxeter, H. S. M. (January 1973), Regular Polytopes (3rd ed.), Dover Publications Inc., p.  12, ISBN   0-486-61480-8
  6. McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, p.  158, ISBN   0-521-81496-0