geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, an apeirogon () or infinite polygon is a generalized

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

with a countably infinite number of sides. Apeirogons are the two-dimensional 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, ''p1m1'', ...

of symmetries.

Definitions

Classical constructive definition

Given a point ''A₀'' in a

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

and a

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

''S'', define the point ''A_i'' to be the point obtained from ''i'' applications of the translation ''S'' to ''A₀'', so ''A_i = Sⁱ(A₀)''. 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 ''E¹'' into infinitely many equal-length segments, generalizing the regular ''n''-gon, which can be defined as a partition of the circle ''S¹'' into finitely many equal-length segments.

Modern abstract definition

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 partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...

''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), one vertex, two vertices (an

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

), 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, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...

formed by the vertices and edges is connected. An abstract polytope is called an abstract

apeirotope In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets. Definition Abstract apeirotope An abstract ''n''-polytope is a partially ordered set ''P'' (whose elements are called ''faces'') such tha ...

if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon. 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

Pseudogon

The regular pseudogon is a partition of the hyperbolic line ''H¹'' (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.

Realizations

Definition

A 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. Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. The classical definition of an apeirogon as an equally-spaced subdivision of the Euclidean line is a realization in this sense, as is the convex subset in the hyperbolic plane formed by the convex hull of equally-spaced points on a horocycle. Other realizations are possible in higher-dimensional spaces.

Symmetries of a realization

The infinite dihedral group ''G'' of symmetries of a realization ''V'' of an abstract apeirogon ''P'' 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.

Moduli space of realizations

Generally, the moduli space of realizations 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 realizations of two-dimensional abstract polytopes (including both polygons and apeirogons), in

s of at most three dimensions, can be classified into six types: *

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

s, * star polygons, *regular apeirogons in the Euclidean line, * infinite skew polygons (infinite zig-zag polygons in the Euclidean plane), *

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s (including star prisms and star antiprisms), and *infinite helical polygons (evenly spaced points along a

helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...

). Abstract apeirogons may be realized in all of these ways, in some cases mapping infinitely many different vertices of an abstract apeirogon onto finitely many points of the realization. An apeirogon also admits star polygon realizations and antiprismatic realizations with a non-discrete set of infinitely many points.

Generalizations

Higher dimension

Apeirohedra are the 3-dimensional analogues of apeirogons, and are the infinite analogues of

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

. More generally, ''n''-

s 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 ...

References

External links

* * {{Polygons Polygons by the number of sides Infinity