View source for Apeirogon

{{Short description|Polygon with an infinite number of sides}}
{{For|the book|Apeirogon (novel)}}
{{Infobox Polygon
| name        = Apeirogon ([[Regular polygon|regular]])
| image       = regular apeirogon.svg
| edges       = ∞
| schläfli    = {∞}
| coxeter     = {{CDD|node_1|infin|node}}
| angle       = 180°
| dual        = Self-dual
}}
[[File:Regular 2-apeirogon.png|260px|thumb|A partition of the Euclidean line into infinitely many equal-length segments can be understood as a regular apeirogon.]]
In [[geometry]], an '''apeirogon''' ({{etymology|grc|{{noitalic|ἄπειρος}} apeiros|infinite, boundless||{{noitalic|γωνία}} gonia|angle}}) or '''infinite polygon''' is a [[polygon]] with an [[Infinity|infinite]] number of sides. Apeirogons are the rank 2 case of [[Apeirotope|infinite polytopes]]. In some literature, the term "apeirogon" may refer only to the [[Regular polygon|regular]] apeirogon, with an [[infinite dihedral group]] of [[symmetry|symmetries]].<ref name="Coxeter"/>

==Definitions==
===Geometric apeirogon===
Given a point ''A''<sub>0</sub> in a [[Euclidean space]] and a [[translation (geometry)|translation]] ''S'', define the point ''A<sub>i</sub>'' to be the point obtained from ''i'' applications of the translation ''S'' to ''A''<sub>0</sub>, so ''A<sub>i</sub>'' = ''S<sup>i</sup>''(''A''<sub>0</sub>). The set of vertices ''A<sub>i</sub>'' with ''i'' any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the '''regular apeirogon''' as defined by [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]].<ref name="Coxeter">{{cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Regular polytopes |publisher=Methuen & Co. Ltd. |location=London |year=1948 |page=45}}</ref>

A '''regular apeirogon''' can be defined as a partition of the Euclidean line ''E''<sup>1</sup> into infinitely many equal-length segments. It generalizes the regular [[polygon|''n''-gon]], which may be defined as a partition of the circle ''S''<sup>1</sup> into ''finitely'' many equal-length segments.<ref name=Johnson>{{cite book |first=Norman W. |last=Johnson |author-link=Norman Johnson (mathematician) |title=Geometries and transformations |year=2018 |chapter=11: Finite Symmetry Groups |page=226 |url= https://books.google.com/books?id=J7NavgAACAAJ |publisher=[[Cambridge University Press]]|isbn=9781107103405 }}</ref>

===Hyperbolic pseudogon===
The '''regular pseudogon''' is a partition of the hyperbolic line ''H''<sup>1</sup> (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.<ref name=Johnson/>

===Abstract apeirogon===
An [[abstract polytope]] is a [[partially ordered set]] ''P'' (whose elements are called ''faces'') with properties modeling those of the inclusions of faces of [[convex polytope]]s. The ''rank'' (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank ''n'' is called an abstract ''n''-polytope.<ref name="McMullen-Schulte"/>{{rp|pp=22–25}}

For abstract polytopes of rank 2, this means that: A) the elements of the partially ordered set are sets of vertices with either zero vertex (the [[empty set]]), one vertex, two vertices (an [[edge (geometry)|edge]]), or the entire vertex set (a two-dimensional face), ordered by inclusion of sets; B) each vertex belongs to exactly two edges; C) the [[undirected graph]] formed by the vertices and edges is connected.<ref name="McMullen-Schulte">{{cite book| last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | title = Abstract Regular Polytopes | edition = 1st | publisher = [[Cambridge University Press]] | isbn = 0-521-81496-0 | date = December 2002 | url-access = registration | url = https://archive.org/details/abstractregularp0000mcmu]}}</ref>{{rp|pp=22–25}}<ref name="McMullen">{{citation|last=McMullen|first=Peter|author-link=Peter McMullen|doi=10.1007/BF01832961|issue=2–3|journal=Aequationes Mathematicae|mr=1268033|pages=223–239|title=Realizations of regular apeirotopes|volume=47|year=1994|s2cid=121616949 }}</ref>{{rp|p=224}}

An abstract polytope is called an abstract [[apeirotope]] if it has infinitely many elements; an abstract 2-apeirotope is called an '''abstract apeirogon'''.<ref name="McMullen-Schulte" />{{rp|p=25}}

A realization of an abstract polytope is a mapping of its vertices to points a geometric space (typically a [[Euclidean space]]).<ref name="McMullen-Schulte"/>{{rp|p=121}} A faithful realization is a realization such that the vertex mapping is [[injective]].<ref name="McMullen-Schulte"/>{{rp|p=122}}<ref group="note">{{harvcoltxt|McMullen|Schulte|2002}} provide a stricter definition, requiring that the induced maps on higher rank faces be injective as well. However a regular polytope is either degenerate in which case it has no faithful realizations, or every vertex-faithful realization is faithful. The apeirogon is not degenerate and thus this condition is sufficient to show its realizations are faithful.</ref> Every geometric apeirogon is a realization of the abstract apeirogon.

==Symmetries==
[[File:Hyperbolic apeirogon example.png|thumb|The [[order-3 apeirogonal tiling]], {∞,3}, fills the hyperbolic plane with apeirogons whose vertices lie along horocyclic paths.]]

The infinite dihedral group ''G'' of symmetries of a regular geometric apeirogon is generated by two reflections, the product of which translates each vertex of ''P'' to the next.<ref name="McMullen-Schulte"/>{{rp|pp=140–141}}<ref name="McMullen"/>{{rp|p=231}} The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection.<ref name="McMullen-Schulte"/>{{rp|p=141}}<ref name="McMullen"/>{{rp|p=231}}

In an abstract polytope, a ''flag'' is a collection of one face of each dimension, all incident to each other (that is, comparable in the partial order); an abstract polytope is called ''regular'' if it has symmetries (structure-preserving permutations of its elements) that take any flag to any other flag. In the case of a two-dimensional abstract polytope, this is automatically true; the symmetries of the apeirogon form the [[infinite dihedral group]].<ref name="McMullen-Schulte" />{{rp|p=31}}

A '''symmetric realization''' of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a [[Euclidean space]]) such that every symmetry of the abstract apeirogon corresponds to an [[isometry]] of the images of the mapping.<ref name="McMullen-Schulte"/>{{rp|p=121}}<ref name="McMullen"/>{{rp|p=225}}

==Moduli space==
Generally, the [[moduli space]] of a faithful realization of an abstract polytope is a [[convex cone]] of infinite dimension.<ref name="McMullen-Schulte"/>{{rp|p=127}}<ref name="McMullen"/>{{rp|pp=229–230}} The realization cone of the abstract apeirogon has uncountably infinite [[algebraic dimension]] and cannot be [[Closed set|closed]] in the [[Euclidean topology]].<ref name="McMullen-Schulte"/>{{rp|p=141}}<ref name="McMullen"/>{{rp|p=232}}

==Classification of Euclidean apeirogons==
The symmetric realization of any regular polygon in Euclidean space of dimension greater than 2 is [[Irreducible representation|reducible]], meaning it can be made as a blend of two lower-dimensional polygons.<ref name="McMullen-Schulte"/> This characterization of the regular polygons naturally characterizes the regular apeirogons as well. The discrete apeirogons are the results of blending the 1-dimensional apeirogon with other polygons.<ref name="McMullen"/>{{rp|p=231}} Since every polygon is a quotient of the apeirogon, the blend of any polygon with an apeirogon produces another apeirogon.<ref name="McMullen-Schulte"/>

In two dimensions the discrete regular apeirogons are the [[Skew apeirogon|infinite zigzag polygon]]s,<ref name="Grünbaum">{{cite journal|last=Grünbaum |first=B.|author-link=Branko Grünbaum |title=Regular polyhedra – old and new |journal=[[Aequationes Mathematicae]] |volume=16 |year=1977 |page=1–20 |doi=10.1007/BF01836414|s2cid=125049930 }}</ref> resulting from the blend of the 1-dimensional apeirogon with the [[digon]], represented with the [[Schläfli symbol]] {{math|{&infin;}#{2{{)}}}}, {{math|{&infin;}#{{(}}{{)}}}}, or <math>\left\{\dfrac{2}{0,1}\right\}</math>.<ref name="McMullen-Schulte"/>

In three dimensions the discrete regular apeirogons are the infinite helical polygons,<ref name="Grünbaum"/> with vertices spaced evenly along a [[helix]].  These are the result of blending the 1-dimensional apeirogon with a 2-dimensional polygon, {{math|{&infin;}#{{{mvar|p}}/{{mvar|q}}{{)}}}} or <math>\left\{\dfrac{p}{0,q}\right\}</math>.<ref name="McMullen-Schulte"/>

==Generalizations==
===Higher rank===
{{Main|Apeirotope|Apeirohedron}}
[[Apeirohedron|Apeirohedra]] are the rank 3 analogues of apeirogons, and are the infinite analogues of [[polyhedron|polyhedra]].<ref>{{cite journal|last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter|title=Regular Skew Polyhedra in Three and Four Dimensions. |journal=Proc. London Math. Soc. |volume=43 |pages=33–62 |year=1937}}</ref> More generally, ''n''-[[apeirotope]]s or infinite ''n''-polytopes are the ''n''-dimensional analogues of apeirogons, and are the infinite analogues of ''n''-[[polytope]]s.<ref name="McMullen-Schulte"/>{{rp|pp=22–25}}

==See also==
*[[Apeirogonal tiling]]
*[[Apeirogonal prism]]
*[[Apeirogonal antiprism]]
*[[Teragon]], a fractal generalized polygon that also has infinitely many sides

==Notes==
{{reflist|group=note}}

==References==
{{Reflist}}

==External links==
*{{mathworld |author=''Russell, Robert A.'' |urlname=Apeirogon |title=Apeirogon}}
*{{GlossaryForHyperspace |anchor=Apeirogon |title=Apeirogon}}

{{Polygons}}

[[Category:Polygons by the number of sides]]
[[Category:Infinity]]