# Apeirogon

In geometry, an apeirogon (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are infinite 2-polytopes and can be considered as limits of an n-sided polygon as n approaches infinity.

The regular apeirogon
Edges and vertices
Schläfli symbol{∞}
Coxeter diagram
Internal angle (degrees)180°
Dual polygonSelf-dual
An apeirogon can be defined as a partition of the Euclidean line into infinitely many equal-length segments.

## Definitions

Hyperbolic pseudogon example

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

Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si A0. The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal segments of a line. This sequence is called the regular apeirogon.[1]

A regular apeirogon can be equivalently be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments, an infinite polygon inscribed in a horocycle, or an infinite polygon inscribed in the absolute circle of the hyperbolic plane.[2]

In general, an apeirogon or infinite polygon may be defined as any infinite 2-polytope.[3][4] In some literature, the term "apeirogon" may refer only to the regular apeirogon.[1]

## Classification of polygons

Regular polygons (equiangular and equilateral 2-polytopes that are not necessarily finite) in the 3-dimensional Euclidean space E3 can be classified into 6 types: convex polygons, star polygons, the regular apeirogon, infinite zig-zag polygons, infinite skew polygons, and infinite helical polygons.[5]

## Generalizations

### Higher dimension

An infinite 3-polytope, the infinite analogue of polyhedra, are the 3-dimensional analogues of apeirogons.[6]

### Hyperbolic space

The regular pseudogon is a partition of the hyperbolic line H1 (instead of the Euclidean line} into segments of length 2λ.[2]