# 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${\displaystyle 0}$
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:

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

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