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Order-2 apeirogonal tiling

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Apeirogonal tiling
Order-2 apeirogonal tiling
Type Regular tiling
Vertex configuration ∞.∞
[[File:|40px]]
Face configuration V2.2.2...
Schläfli symbol(s) {∞,2}
Wythoff symbol(s) 2 | ∞ 2
2 2 | ∞
Coxeter diagram(s)
Symmetry [∞,2], (*∞22)
Rotation symmetry [∞,2]+, (∞22)
Dual Apeirogonal hosohedron
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron[1] is a tessellation (gap-free filling with repeated shapes) of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {∞, 2}. Two apeirogons, joined along all their edges, can completely fill the entire plane as an apeirogon is infinite in size and has an interior angle of 180°, which is half of a full 360°.

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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

See also

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Notes

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References

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  1. ^ Conway (2008), p. 263
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5
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