Order-2 apeirogonal tiling

Apeirogonal tiling
Type	Regular tiling
Vertex configuration	∞.∞
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 or apeirogonal dihedron is a tiling 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°.

Related tilings and polyhedra

The apeirogonal tiling is the arithmetic limit of the family of dihedra {p, 2}, as p tends to infinity, thereby turning the dihedron 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.∞

See also

• Order-3 apeirogonal tiling - hyperbolic tiling
• Order-4 apeirogonal tiling - hyperbolic tiling

Notes

References

• Jim McNeill: Tessellations of the Plane

Tessellation & tiling   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & tessellation Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagon Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet   Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring 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 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82