|Bowers style acronym||Aze|
|Coxeter diagram||x∞o ()|
|Vertex figure||Dyad, length 2|
|Measures (edge length 1)|
|Abstract & topological properties|
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.
Coordinates for the vertices of a regular apeirogon of edge length 1 are given by
where i ranges over all of the integers.
A regular apeirogon has the following Coxeter diagrams:
- x∞o (regular)
- x∞x (two alternating edge types)
- s∞o (as alternation)
- Klitzing, Richard. "aze".
- Wikipedia Contributors. "Apeirogon".