Apeirogonal prism

Apeirogonal prism
Apeirogonal prism
Type Semiregular tiling
Vertex configuration
4.4.∞
Schläfli symbol t{2,∞}
Wythoff symbol 2 ∞ | 2
Coxeter diagram
Symmetry [∞,2], (*∞22)
Rotation symmetry [∞,2]+, (∞22)
Bowers acronym Azip
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]

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]

  • Uniform variant with alternate colored square faces.
    Uniform variant with alternate colored square faces.
  • Its dual tiling is an apeirogonal bipyramid.
    Its dual tiling is an apeirogonal bipyramid.

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.

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 image Tiling name
Parent 2 | ∞ 2 {∞,2} ∞.∞ Apeirogonal
dihedron
Truncated 2 2 | ∞ t{∞,2} 2.∞.∞
Rectified 2 | ∞ 2 r{∞,2} 2.∞.2.∞
Birectified
(dual) 		∞ | 2 2 {2,∞} 2∞ Apeirogonal
hosohedron
Bitruncated 2 ∞ | 2 t{2,∞} 4.4.∞ Apeirogonal
prism
Cantellated ∞ 2 | 2 rr{∞,2}
Omnitruncated
(Cantitruncated) 		∞ 2 2 | tr{∞,2} 4.4.∞
Snub | ∞ 2 2 sr{∞,2} 3.3.3.∞ Apeirogonal
antiprism

Notes

  1. ^ Conway (2008), p.263

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
  • Conway, John H.; Heidi Burgiel; Chaim Goodman-Strauss (2008). The Symmetries of Things. ISBN 978-1-56881-220-5.


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