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	$0$
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	W2×A1
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

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.

External links

• Klitzing, Richard. "azip".

• Wikipedia Contributors. "Apeirogonal prism".