Infinite-order triangular tiling

Infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex figure	3∞
Schläfli symbol	{3,∞}
Wythoff symbol	∞ | 3 2
Coxeter diagram
Symmetry group	[∞,3], (*∞32)
Dual	Order-3 apeirogonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

The {3,3,∞} honeycomb has {3,∞} vertex figures.

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.

Symmetry

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

Alternated colored tiling

*∞∞∞ symmetry

Apollonian gasket with *∞∞∞ symmetry

Related polyhedra and tiling

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

*n32 symmetry mutation of regular tilings: {3,n}
Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
3.3	33	34	35	36	37	38	3∞	312i	39i	36i	33i

Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32)							[∞,3]+ (∞32)	[1+,∞,3] (*∞33)		[∞,3+] (3*∞)
		=	=	=				= or	= or	=
{∞,3}	t{∞,3}	r{∞,3}	t{3,∞}	{3,∞}	rr{∞,3}	tr{∞,3}	sr{∞,3}	h{∞,3}	h2{∞,3}	s{3,∞}
Uniform duals
V∞3	V3.∞.∞	V(3.∞)2	V6.6.∞	V3∞	V4.3.4.∞	V4.6.∞	V3.3.3.3.∞	V(3.∞)3		V3.3.3.3.3.∞

Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33)							[(∞,3,3)]+, (∞33)
(∞,∞,3)	t0,1(∞,3,3)	t1(∞,3,3)	t1,2(∞,3,3)	t2(∞,3,3)	t0,2(∞,3,3)	t0,1,2(∞,3,3)	s(∞,3,3)
Dual tilings
V(3.∞)3	V3.∞.3.∞	V(3.∞)3	V3.6.∞.6	V(3.3)∞	V3.6.∞.6	V6.6.∞	V3.3.3.3.3.∞

Other infinite-order triangular tilings

A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

See also

Wikimedia Commons has media related to Infinite-order triangular tiling.

• Infinite-order tetrahedral honeycomb
• List of regular polytopes
• List of uniform planar tilings
• Tilings of regular polygons
• Triangular tiling
• Uniform tilings in hyperbolic plane

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. 

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld. 

• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld. 

Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & 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 & 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