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In geometry, an apeirogonal hosohedron or infinite hosohedron[1] is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {2,∞}.

Apeirogonal hosohedron
Apeirogonal hosohedron
Type Regular tiling
Vertex configuration 2
[[File:|40px]]
Face configuration V∞2
Schläfli symbol(s) {2,∞}
Wythoff symbol(s) ∞ | 2 2
Coxeter diagram(s) CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.png
Symmetry [∞,2], (*∞22)
Rotation symmetry [∞,2]+, (∞22)
Dual Order-2 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

Related tilings and polyhedra

The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra {2,p}, as p tends to infinity, thereby turning the hosohedron into a Euclidean tiling. All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges.

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 apeirogonal tilings
(∞ 2 2) Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff 2 | ∞ 2 2 2 | ∞ 2 | ∞ 2 2 ∞ | 2 ∞ | 2 2 ∞ 2 | 2 ∞ 2 2 | | ∞ 2 2
Schläfli {∞,2} t{∞,2} r{∞,2} t{2,∞} {2,∞} rr{∞,2} tr{∞,2} sr{∞,2}
Coxeter CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2x.pngCDel node 1.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.png
Image
Vertex figure
Apeirogonal tiling.svg
{∞,2}
Apeirogonal tiling.svg
∞.∞
Apeirogonal tiling.svg
∞.∞
Infinite prism.svg
4.4.∞
Apeirogonal hosohedron.svg
{2,∞}
Infinite prism.svg
4.4.∞
Infinite prism alternating.svg
4.4.∞
Infinite antiprism.svg
3.3.3.∞

Notes

Conway (2008), p. 263

References

The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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