Apeirogonal prism

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Apeirogonal prism
Infinite prism.svg
Type Semiregular tiling
Vertex configuration Infinite prism verf.svg
4.4.
Schläfli symbol t{2,}
Wythoff symbol 2 | 2
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Symmetry [,2], (*22)
Rotation symmetry[,2]+, (22)
Bowers acronymAzip
Dual Apeirogonal bipyramid
Properties Vertex-transitive

In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane. [1]

Contents

Thorold Gosset called it a 2-dimensional semi-check, like a single row of a checkerboard.[ citation needed ]

If the sides are squares, it is a uniform tiling. If colored with two sets of alternating squares it is still uniform.[ citation needed ]

The apeirogonal tiling is the arithmetic limit of the family of prisms t{2, p} or p.4.4, as p tends to infinity, thereby turning the prism into a Euclidean tiling.

An alternation operation can create an apeirogonal antiprism composed of three triangles and one apeirogon at each vertex.

Infinite antiprism.svg

Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism.

Order-2 regular or uniform apeirogonal tilings
(∞ 2 2) Wythoff
symbol
Schläfli
symbol
Coxeter
diagram
Vertex
config.
Tiling imageTiling name
Parent2 | ∞ 2{∞,2}CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png∞.∞ Apeirogonal tiling.svg Apeirogonal
dihedron
Truncated2 2 |t{∞,2}CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node.png2.∞.∞
Rectified2 | ∞ 2r{∞,2}CDel node.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node.png2.∞.2.∞
Birectified
(dual)
| 2 2{2,∞}CDel node.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node 1.png2 Apeirogonal hosohedron.svg Apeirogonal
hosohedron
Bitruncated2 ∞ | 2t{2,∞}CDel node.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node 1.png4.4.∞ Infinite prism.svg Apeirogonal
prism
Cantellated∞ 2 | 2rr{∞,2}CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node 1.png
Omnitruncated
(Cantitruncated)
∞ 2 2 |tr{∞,2}CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node 1.png4.4.∞ Infinite prism alternating.svg
Snub| ∞ 2 2sr{∞,2}CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.png3.3.3.∞ Infinite antiprism.svg Apeirogonal
antiprism

Notes

  1. Conway (2008), p.263

References