Apeirogonal hosohedron

Geometric tiling of the plane

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) |
Symmetry | [∞,2], (*∞22)
Rotation symmetry | [∞,2]+, (∞22)
Dual | Order-2 apeirogonal tiling
Properties | Vertex-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[edit]

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 regular or uniform apeirogonal tilings  (∞ 2 2)  | Parent  | Truncated  | Rectified  | Bitruncated  | Birectified
(dual)  | Cantellated  | Omnitruncated
(Cantitruncated)  | Snub

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Wythoff symbol | 2 | ∞ 2  | 2 2 | ∞  | 2 | ∞ 2  | 2 ∞ | 2  | ∞ | 2 2  | ∞ 2 | 2  | ∞ 2 2 | | | ∞ 2 2
Schläfli symbol | {∞,2}  | t{∞,2}  | r{∞,2}  | t{2,∞}  | {2,∞}  | rr{∞,2}  | tr{∞,2}  | sr{∞,2}
Coxeter-Dynkin diagram |  |  |  |  |  |  |  |
Vertex config. | ∞.∞  | ∞.∞  | ∞.∞  | 4.4.∞  | 2∞ | 4.4.∞  | 4.4.∞  | 3.3.3.∞
Tiling image  |  |  |  |  |  |  |  |
Tiling name  | Apeirogonal dihedron | Apeirogonal dihedron  | Apeirogonal dihedron  | Apeirogonal prism | Apeirogonal hosohedron | Apeirogonal prism  | Apeirogonal prism  | Apeirogonal antiprism

## Notes[edit]

1. ^ Conway (2008), p. 263

## References[edit]

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

## External links[edit]

* Jim McNeill: Tessellations of the Plane

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Tessellation

Periodic

* Pythagorean
* Rhombille
* Schwarz triangle
* Rectangle
* Domino
* Uniform tiling and honeycomb
* Coloring
* Convex
* Kisrhombille
* Wallpaper group
* Wythoff

Aperiodic

* Ammann–Beenker
* Aperiodic set of prototiles
* List
* Einstein problem
* Socolar–Taylor
* Gilbert
* Penrose
* Pentagonal
* Pinwheel
* Quaquaversal
* Rep-tile and Self-tiling
* Sphinx
* Socolar
* Truchet

Other

* Anisohedral and Isohedral
* Architectonic and catoptric
* Circle Limit III
* Computer graphics
* Honeycomb
* Isotoxal
* List
* Packing
* Problems
* Domino
* Wang
* Heesch's
* Squaring
* Dividing a square into similar rectangles
* Prototile
* Conway criterion
* Girih
* Regular Division of the Plane
* Regular grid
* Substitution
* Voronoi
* Voderberg

By vertex type

| Spherical|

* 2n
* 33.n
* V33.n
* 42.n
* V42.n

|

Regular|

* 2∞
* 36
* 44
* 63

Semi-
regular|

* 32.4.3.4
* V32.4.3.4
* 33.42
* 33.∞
* 34.6
* V34.6
* 3.4.6.4
* (3.6)2
* 3.122
* 42.∞
* 4.6.12
* 4.82

Hyper-
bolic|

* 32.4.3.5
* 32.4.3.6
* 32.4.3.7
* 32.4.3.8
* 32.4.3.∞
* 32.5.3.5
* 32.5.3.6
* 32.6.3.6
* 32.6.3.8
* 32.7.3.7
* 32.8.3.8
* 33.4.3.4
* 32.∞.3.∞
* 34.7
* 34.8
* 34.∞
* 35.4
* 37
* 38
* 3∞
* (3.4)3
* (3.4)4
* 3.4.62.4
* 3.4.7.4
* 3.4.8.4
* 3.4.∞.4
* 3.6.4.6
* (3.7)2
* (3.8)2
* 3.142
* 3.162
* (3.∞)2
* 3.∞2
* 42.5.4
* 42.6.4
* 42.7.4
* 42.8.4
* 42.∞.4
* 45
* 46
* 47
* 48
* 4∞
* (4.5)2
* (4.6)2
* 4.6.12
* 4.6.14
* V4.6.14
* 4.6.16
* V4.6.16
* 4.6.∞
* (4.7)2
* (4.8)2
* 4.8.10
* V4.8.10
* 4.8.12
* 4.8.14
* 4.8.16
* 4.8.∞
* 4.102
* 4.10.12
* 4.122
* 4.12.16
* 4.142
* 4.162
* 4.∞2
* (4.∞)2
* 54
* 55
* 56
* 5∞
* 5.4.6.4
* (5.6)2
* 5.82
* 5.102
* 5.122
* (5.∞)2
* 64
* 65
* 66
* 68
* 6.4.8.4
* (6.8)2
* 6.82
* 6.102
* 6.122
* 6.162
* 73
* 74
* 77
* 7.62
* 7.82
* 7.142
* 83
* 84
* 86
* 88
* 8.62
* 8.122
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

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