Apeirogon

The apeirogon, or aze, is the only regular tiling of 1-dimensional space. It consists of an infinite sequence of dyads. It can be thought of as an infinite-sided regular polygon.

The apeirogon can be seen as the 1D hypercubic tiling.

In Euclidean space, all regular apeirogons have all vertices inscribed in a line. However, in hyperbolic space, there are three types: one can be inscribed in a horocycle (normally called an apeirogon), one can be inscribed in a hypercycle (called a pseudogon) and one can be inscribed in a line. Pseudogons form a continuum of sizes – there's not a single family of pseudogons up to similarity. Apeirogons can tile the hyperbolic plane, as in the apeirogonal tiling.

Vertex coordinates
Coordinates for the vertices of a regular apeirogon of edge length 1 are given by where i ranges over all of the integers.
 * (i),

Representations
A regular apeirogon has the following Coxeter diagrams:


 * x∞o (regular)
 * x∞x (two alternating edge types)
 * s∞o (as alternation)

Variations
Besides the regular apeirogon, non-regular apeirogons can be considered to exist by dividing the line into segments of non-equal length. One such non-regular apeirogon, which can be called a diapeirogon or truncated apeirogon, has edges of two alternating lengths, and remains isogonal.