Apeirogon

Apeirogon
Rank	2
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Aze
Coxeter diagram	x∞o ()
Schläfli symbol	{∞}
Elements
Edges	N
Vertices	N
Vertex figure	Dyad, length 2
Measures (edge length 1)
Vertex density	1
Related polytopes
Army	Aze
Dual	Apeirogon
Conjugate	None
Abstract & topological properties
Flag count	2N
Orientable	Yes
Properties
Symmetry	W2, order 2N
Convex	Yes

For a general polytope with infinitely many facets, see apeirotope.

Apeirogon inscribed in a horocycle, a curve in hyperbolic space that approaches a single ideal point.

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

• (i),

where i ranges over all of the integers.

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.

External links

• Klitzing, Richard. "aze".

• Wikipedia Contributors. "Apeirogon".