Apeirogonal hosohedron

Apeirogonal hosohedron

TypeRegular tiling
Vertex configuration2
Schläfli symbol(s){2,}
Wythoff symbol(s) | 2 2
Coxeter diagram(s)
Symmetry[,2], (*22)
Rotation symmetry[,2]+, (22)
DualOrder-2 apeirogonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

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,∞}.

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.

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.

(∞ 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 t0{∞,2} t0,1{∞,2} t1{∞,2} t1,2{∞,2} t2{∞,2} t0,2{∞,2} t0,1,2{∞,2} s{∞,2}
Coxeter
Image
Vertex figure

{∞,2}

∞.∞

∞.∞

4.4.∞

{2,∞}

4.4.∞

4.4.∞

3.3.3.∞

Notes

  1. Conway (2008), p. 263

References

External links

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