Apeirogonal prism

From Polytope Wiki
Jump to navigation Jump to search
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
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[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]