Infinite-order triangular tiling

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Template:Short description

Infinite-order triangular tiling
Infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3∞
Schläfli symbol {3,∞}
Wythoff symbol ∞ | 3 2
Coxeter diagram Template:CDD
Template:CDD
Symmetry group [∞,3], (*∞32)
Dual Order-3 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive
File:H3 33inf UHS plane at infinity.png
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)}, Template:CDD. 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.

File:Infinite-order triangular tiling.svg
Alternated colored tiling 		File:Iii symmetry mirrors.png
*∞∞∞ symmetry 		File:Apolleangasket symmetry.png
Apollonian gasket with *∞∞∞ symmetry

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

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Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
File:Trigonal dihedron.svg File:Uniform tiling 332-t2.svg File:Uniform tiling 432-t2.svg File:Uniform tiling 532-t2.svg File:Uniform polyhedron-63-t2.svg File:Order-7 triangular tiling.svg File:H2-8-3-primal.svg File:H2 tiling 23i-4.png File:H2 tiling 23j12-4.png File:H2 tiling 23j9-4.png File:H2 tiling 23j6-4.png File:H2 tiling 23j3-4.png
3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i
Paracompact uniform tilings in [∞,3] family Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32) 		[1+,∞,3]
(*∞33) 		[∞,3+]
(3*∞)
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File:H2-I-3-dual.svg File:H2 tiling 23i-3.png File:H2 tiling 23i-2.png File:H2 tiling 23i-6.png File:H2 tiling 23i-4.png File:H2 tiling 23i-5.png File:H2 tiling 23i-7.png File:Uniform tiling i32-snub.png File:H2 tiling 33i-1.png File:H2 snub 33ia.png
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
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File:H2 tiling 23i-4.png File:Ord-infin triakis triang til.png File:Ord3infin qreg rhombic til.png File:H2checkers 33i.png File:H2-I-3-dual.svg File:Deltoidal triapeirogonal til.png File:H2checkers 23i.png File:Order-3-infinite floret pentagonal tiling.png File:Alternate order-3 apeirogonal tiling.png
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 Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Symmetry: [(∞,3,3)], (*∞33) [(∞,3,3)]+, (∞33)
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File:H2 tiling 33i-1.png File:H2 tiling 33i-3.png File:H2 tiling 33i-2.png File:H2 tiling 33i-6.png File:H2 tiling 33i-4.png File:H2 tiling 33i-5.png File:H2 tiling 33i-7.png File:H2 snub 33ia.png
(∞,∞,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
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File:Ord3infin qreg rhombic til.png File:H2checkers 33i.png
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:

File:Ideal-triangle hyperbolic tiling.svg

See also

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References

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