Apeirogon

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Apeirogon
Rank2
Dimension1
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymAze
Coxeter diagramx∞o ()
Schläfli symbol{∞}
Elements
EdgesN
VerticesN
Vertex figureDyad, length 2
Measures (edge length 1)
Vertex density1
Related polytopes
ArmyAze
DualApeirogon
ConjugateNone
Abstract & topological properties
Flag count2N
OrientableYes
Properties
SymmetryW2, order 2N
ConvexYes
NatureTame
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 belongs to a number of notable families, it is a hypercubic honeycomb, an apeir simplex, an apeir orthoplex, and an apeir hypercube.

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[edit | edit source]

Coordinates for the vertices of a regular apeirogon of edge length 1 are given by

  • ,

where i  ranges over all of the integers.

Representations[edit | edit source]

A regular apeirogon has the following Coxeter diagrams:

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

Variations[edit | edit source]

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[edit | edit source]