# Apeirogonal prism

Apeirogonal prism Rank3
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymAzip
Coxeter diagramx2x∞o (     )
Elements
FacesN squares, 2 apeirogons
EdgesN+2N
Vertices2N
Vertex figureIsosceles triangle, edge lengths 2, 2, 2
Measures (edge length 1)
Height1
Vertex density$0$ Level of complexity3
Related polytopes
ArmyAzip
RegimentAzip
DualApeirogonal tegum
ConjugateNone
Abstract & topological properties
Flag count12N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryW2×A1
ConvexYes
NatureTame

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

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

• $\left(\pm\tfrac12,\,i,\,0\right)$ .

## Representations

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

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 $a$ and side edges of length $b$ can be alternated to form an apeirogonal antiprism with base edges of length $2a$ and side edges of lengths $\sqrt{a^2+b^2}$ . In particular if the side edges are $\tfrac{\sqrt3}{2}$ times the length of the base edges this gives a uniform apeirogonal antiprism.