# Apeirogonal prism

Apeirogonal prism | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Euclidean |

Notation | |

Bowers style acronym | Azip |

Coxeter diagram | x2x∞o () |

Elements | |

Faces | N squares, 2 apeirogons |

Edges | N+2N |

Vertices | 2N |

Vertex figure | Isosceles triangle, edge lengths 2, √2, √2 |

Measures (edge length 1) | |

Height | 1 |

Vertex density | |

Level of complexity | 3 |

Related polytopes | |

Army | Azip |

Regiment | Azip |

Dual | Apeirogonal tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 12N |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | W_{2}×A_{1} |

Convex | Yes |

Nature | Tame |

The **apeirogonal prism**, or **azip**, is a prismatic uniform tiling of the Euclidean plane. It consists of 2 apeirogons and countably many squares. Each vertex joins one apeirogon and two squares. As the name suggests, it is a prism based on the apeirogon.

## Vertex coordinates[edit | edit source]

An apeirogonal prism of edge length 1 has vertex coordinates given by, where i ranges over the integers:

- .

## Representations[edit | edit source]

An apeirogonal prism has the following Coxeter diagrams:

- x2x∞o () (full symmetry)
- x2x∞x () (as diapeirogonal prism, squares of two types)
- s2s∞x () (as diapeirogonal trapezoprism)
- xx∞oo&#x (apeirogonal frustum)
- xx∞xx&#x (diapeirogonal frustum)

## Semi-uniform variant[edit | edit source]

The apeirogonal prism has a semi-uniform variant of the form x y∞o that maintains its full symmetry. This variant uses rectangles as its sides.

An apeirogonal prism with base edges of length and side edges of length can be alternated to form an apeirogonal antiprism with base edges of length and side edges of lengths . In particular if the side edges are times the length of the base edges this gives a uniform apeirogonal antiprism.

## External links[edit | edit source]

- Klitzing, Richard. "azip".

- Wikipedia Contributors. "Apeirogonal prism".