# Apeirogon

Apeirogon | |
---|---|

Rank | 2 |

Dimension | 1 |

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 | W_{2}, order 2N |

Convex | Yes |

Nature | Tame |

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]

- Klitzing, Richard. "aze".
- Wikipedia contributors. "Apeirogon".