Apeirogon

Regular apeirogon
Edges and vertices	∞
Schläfli symbol	{∞}
Coxeter diagram
Dual polygon	Self-dual

In geometry, an apeirogon (from the Greek word "ἄπειρος" apeiros, meaning infinite, boundless and "γωνία" gonia, meaning angle) is a generalized polygon with a countably infinite number of sides. It can be considered as the limit of an n-sided polygon as n approaches infinity. The interior of a linear apeirogon can be defined by a direction order of vertices, and defining half the plane as the interior.

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

A regular apeirogon has equal edge lengths, just like any regular polygon, {p}. Its Schläfli symbol is {∞}, and its Coxeter-Dynkin diagram is . 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. Dually 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.

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