geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an apeirogon () or infinite polygon is a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an

infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p''1''m'' ...

of symmetries.

Definitions

Geometric apeirogon

Given a point ''A''₀ in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

and a

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

''S'', define the point ''A_i'' to be the point obtained from ''i'' applications of the translation ''S'' to ''A''₀, so ''A_i'' = ''Sⁱ''(''A''₀). The set of vertices ''A_i'' 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 H. S. M. Coxeter. A regular apeirogon can be defined as a partition of the Euclidean line ''E''¹ into infinitely many equal-length segments. It generalizes the regular ''n''-gon, which may be defined as a partition of the circle ''S''¹ into ''finitely'' many equal-length segments.

Hyperbolic pseudogon

The regular pseudogon is a partition of the hyperbolic line ''H''¹ (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.

Abstract apeirogon

abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

is a

partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

''P'' (whose elements are called ''faces'') with properties modeling those of the inclusions of faces of convex polytopes. 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. 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 In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

), one vertex, two vertices (an 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 In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...

formed by the vertices and edges is connected. An abstract polytope is called an abstract apeirotope if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon. A realization of an abstract polytope is a mapping of its vertices to points a geometric space (typically a

). A faithful realization is a realization such that the vertex mapping is

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

. 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. Every geometric apeirogon is a realization of the abstract apeirogon.

Symmetries

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. 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. 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

. A symmetric realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a

) such that every symmetry of the abstract apeirogon corresponds to an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

of the images of the mapping.

Moduli space

Generally, the

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...

of a faithful realization of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...

Classification of Euclidean apeirogons

The symmetric realization of any regular polygon in Euclidean space of dimension greater than 2 is reducible, meaning it can be made as a blend of two lower-dimensional polygons. 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. Since every polygon is a quotient of the apeirogon, the blend of any polygon with an apeirogon produces another apeirogon. In two dimensions the discrete regular apeirogons are the infinite zigzag polygons, resulting from the blend of the 1-dimensional apeirogon with the digon, represented with the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

, , or

\left\

. In three dimensions the discrete regular apeirogons are the infinite helical polygons, with vertices spaced evenly along a helix. These are the result of blending the 1-dimensional apeirogon with a 2-dimensional polygon, {{math, {∞}#{{{mvar, p/{{mvar, q{{) or

\left\{\dfrac{p}{0,q}\right\}

Generalizations

Higher rank

{{Main, Apeirotope, Apeirohedron Apeirohedra are the rank 3 analogues of apeirogons, and are the infinite analogues of

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

.{{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 More generally, ''n''- apeirotopes or infinite ''n''-polytopes are the ''n''-dimensional analogues of apeirogons, and are the infinite analogues of ''n''-

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s.{{rp, pp=22–25

Notes

{{reflist, group=note

References

{{Reflist

External links

*{{mathworld , author=''Russell, Robert A.'' , urlname=Apeirogon , title=Apeirogon *{{GlossaryForHyperspace , anchor=Apeirogon , title=Apeirogon {{Polygons Polygons by the number of sides Infinity