Apeirogon

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Regular apeirogon
Regular apeirogon.png
Edges and vertices
Schläfli symbol {∞}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.png
Dual polygon Self-dual

In geometry, an apeirogon is a generalized polygon with a countably infinite number of sides.[1] It can be considered as the limit of a n-sided polygon as n approaches infinity.

This article describes an aperiogon in its linear form as a tessellation or partition of a line.

See also: Skew apeirogon

Regular apeirogon[edit]

A regular apeirogon has equal edge lengths, just like any regular polygon, {p}. Its Schläfli symbol is {∞}, and Coxeter-Dynkin diagram CDel node 1.pngCDel infin.pngCDel node.png. It is the first in the dimensional family of regular hypercubic honeycombs.

This line may be considered as a circle of infinite radius, by analogy with regular polygons with great number of edges, which resemble a circle.

In two dimensions, a regular apeirogon divides the plane into two half-planes as a regular apeirogonal dihedron. The interior of an apeirogon can be defined by its orientation, filling one half plane. Alternately the apeirogonal hosohedron has digon faces and an apeirogonal vertex figure, {2, ∞}. A truncated apeirogonal hosohedron becomes a apeirogonal prism, with each vertex bounded by two squares and an apeirogon. An alternated apeirogonal prism is a apeirogonal antiprism, with each vertex bounded by three triangles and an apeirogon.

Euclidean tilings
Regular Uniform
∞.∞ 2 4.4.∞ 3.3.3.∞
Apeirogonal tiling.png Apeirogonal hosohedron.png Infinite prism tiling.png Infinite antiprism.png
{∞, 2}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png
{2, ∞}
CDel node 1.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
t{2, ∞}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
sr{2, ∞}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.png

The regular aperigon can also be seen within the edges of 4 of the regular, uniform tilings, and 5 of the uniform dual in the Euclidean plane.

3 directions 1 direction 2 directions
Tiling Semiregular 3-6-3-6 Trihexagonal.svg
Hexadeltille
Tiling Regular 3-6 Triangular.svg
Deltile
Tiling Semiregular 3-3-3-4-4 Elongated Triangular.svg
Isosnub quadrille
Tiling Regular 4-4 Square.svg
Quadrille
3 directions 6 directions 1 direction 4 directions
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
Tetrille
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
Kisdeltile
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg
Kisrhombille
Tiling Dual Semiregular V3-3-3-4-4 Prismatic Pentagonal.svg
iso(4-)pentille
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
Kisquadrille

Irregular apeirogon[edit]

An isogonal apeirogon have a single type of vertex and alternates two types of edges.

A quasiregular apeirogon is an isogonal apeirogon with equal edge lengths.

An isotoxal apeirogon, being the dual of an isogonal one, has one type of edge, and two types of vertices, and therefore geometrically identical to the regular apeirogon. It can be shown seen by drawing vertices in alternate colors.

All of these will have half the symmetry (double the fundamental domain sizes) of the regular apeirogon.

Regular ... Regular apeirogon.png ...
Quasiregular ... Uniform apeirogon.png ...
Isogonal ... Isogonal apeirogon linear.png ...
Isotoxal ... Isotoxal linear apeirogon.png ...

Apeirogons in hyperbolic plane[edit]

A regular pseudogon, {iπ/λ}, the Poincaré disk model, with perpendicular reflection lines shown, separated by length λ.

Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.

Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.

Norman Johnson calls the general apeirogon (divergent mirror form) a pseudogon, circumscribed by a hypercycle, with and regular pseudogons as {iπ/λ}, where λ is the periodic distance between the divergent perpendicular mirrors.[2]

The regular tiling {∞, 3} has regular apeirogon faces. Hypercyclic apeirogons can also be isogonal or quasiregular, with truncated apeirogon faces, t{∞}, like the tiling tr{∞,3}, with two types of edges, alternately connecting to triangles or other apeirogons.

Regular and uniform tilings with apeirogons
{∞, 3} tr{∞, 3} tr{iπ/λ,3}
Hyperbolic apeirogon example.png
Regular: {∞}
H2 tiling 23i-7.png
Quasiregular: t{∞}
H2 tiling 23j12-7.png
Quasiregular: t{iπ/λ}

See also[edit]

References[edit]

  1. ^ Coxeter, Regular polytopes, p.45
  2. ^ Norman Johnson, Geometries and symmetries, (2015), Chapter 11. Finite symmetry groups, Section 11.2 The polygonal groups. p.141
  • Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed. ed.). New York: Dover Publications. pp. 121–122. ISBN 0-486-61480-8. 
  • Grünbaum, B. Regular polyhedra - old and new, Aequationes Math. 16 (1977) p. 1-20 [1]
  • Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.  (1st ed, 1957) 5.2 The Petrie polygon {p,q}.

External links[edit]