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Apeirogonal prism

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(Redirected from Apeirogonal bipyramid) Prism with an infinite-sided polygon base
Apeirogonal prism
Apeirogonal prism
Type Semiregular tiling
Vertex configuration
4.4.∞
Schläfli symbol t{2,∞}
Wythoff symbol 2 ∞ | 2
Coxeter diagram
Symmetry , (*∞22)
Rotation symmetry , (∞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.

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

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

  • 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.

Related tilings and polyhedra

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

Tessellation
Periodic


Aperiodic
Other
By vertex type
Spherical
Regular
Semi-
regular
Hyper-
bolic


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