|The regular apeirogon|
|Edges and vertices||∞|
|Internal angle (degrees)||180°|
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.
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.
A regular apeirogon can be defined equivalently 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.
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.
- Coxeter, H. S. M. (1948). Regular polytopes. London: Methuen & Co. Ltd. p. 45.
- Johnson, N. (2015). "11". Geometries and symmetries. p. 226.
- McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes (1st ed.). Cambridge University Press. p. 25. ISBN 0-521-81496-0.
- "The regular apeirogon" is not the only apeirogon that is regular in the sense of being equiangular and equilateral.
- Grünbaum, B. (1977). "Regular polyhedra – old and new". Aequationes Mathematicae. 16 (1–2): 119. doi:10.1007/BF01836414.
- Coxeter, H. S. M. (1937). "Regular Skew Polyhedra in Three and Four Dimensions". Proc. London Math. Soc. 43: 33–62.