Apeirogonal prism

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:
 * $$\left(±\tfrac12,\,i,\,0\right)$$

Representations
An apeirogonal prism has the following Coxeter diagrams:


 * x x∞o (full symmetry)
 * x x∞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.