Truncated order-3 apeirogonal tiling

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Truncated order-3 apeirogonal tiling
Truncated order-3 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.∞.∞
Schläfli symbol t{∞,3}
Wythoff symbol 2 3 | ∞
Coxeter diagram File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Symmetry group [∞,3], (*∞32)
Dual Infinite-order triakis triangular tiling
Properties Vertex-transitive

In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}.

Dual tiling

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The dual tiling, the infinite-order triakis triangular tiling, has face configuration V3.∞.∞.

File:Ord-infin triakis triang til.png
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This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3] 		*332
[3,3] 		*432
[4,3] 		*532
[5,3] 		*632
[6,3] 		*732
[7,3] 		*832
[8,3]... 		*∞32
[∞,3] 		[12i,3] [9i,3] [6i,3]
Truncated
figures 		File:Spherical triangular prism.svg File:Uniform tiling 332-t01-1-.svg File:Uniform tiling 432-t01.svg File:Uniform tiling 532-t01.svg File:Uniform tiling 63-t01.svg File:Truncated heptagonal tiling.svg File:H2-8-3-trunc-dual.svg File:H2 tiling 23i-3.png File:H2 tiling 23j12-3.png File:H2 tiling 23j9-3.png File:H2 tiling 23j6-3.png
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3}
Triakis
figures 		File:Spherical trigonal bipyramid.svg File:Spherical triakis tetrahedron.svg File:Spherical triakis octahedron.svg File:Spherical triakis icosahedron.svg File:Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg File:Order-7 triakis triangular tiling.svg File:H2-8-3-kis-primal.svg File:Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32) 		[1+,∞,3]
(*∞33) 		[∞,3+]
(3*∞)
File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node h.pngFile:CDel infin.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png File:CDel node h1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel infin.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
File:CDel node h0.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
= File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.png 		File:CDel node h0.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node 1.png 		File:CDel node h0.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
= File:CDel labelinfin.pngFile:CDel branch.pngFile:CDel split2.pngFile:CDel node 1.png 		File:CDel node 1.pngFile:CDel infin.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png File:CDel node h1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png =
File:CDel labelinfin.pngFile:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node.png or File:CDel labelinfin.pngFile:CDel branch 01rd.pngFile:CDel split2.pngFile:CDel node.png 		File:CDel node h1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png =
File:CDel labelinfin.pngFile:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node 1.png or File:CDel labelinfin.pngFile:CDel branch 01rd.pngFile:CDel split2.pngFile:CDel node 1.png 		File:CDel node h0.pngFile:CDel infin.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
= File:CDel labelinfin.pngFile:CDel branch hh.pngFile:CDel split2.pngFile:CDel node h.png
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
File:CDel node f1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node f1.pngFile:CDel infin.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel infin.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel infin.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngFile:CDel infin.pngFile:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node fh.pngFile:CDel infin.pngFile:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png File:CDel node fh.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel infin.pngFile:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png
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.∞

See also

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Wikimedia Commons has media related to Uniform tiling 3-i-i.

References

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  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
[edit | edit source]
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
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