Truncated octagonal tiling

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Truncated octagonal tiling
Truncated octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.16.16
Schläfli symbol t{8,3}
Wythoff symbol 2 3 | 8
Coxeter diagram Error creating thumbnail: File:CDel 3.pngFile:CDel node.png
Symmetry group [8,3], (*832)
Dual Order-8 triakis triangular tiling
Properties Vertex-transitive

In geometry, the truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.

Dual tiling

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The dual tiling has face configuration V3.16.16.

File:H2-8-3-kis-primal.svg
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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 Error creating thumbnail: 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.∞.∞

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832) 		[1+,8,3]
(*443) 		[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8} 		tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
Error creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node.png Error creating thumbnail: File:CDel 3.pngFile:CDel node.png File:CDel node.pngError creating thumbnail: File:CDel 3.pngFile:CDel node.png File:CDel node.pngError creating thumbnail: File:CDel 3.png File:CDel node.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.png Error creating thumbnail: File:CDel node.pngFile:CDel 3.png Error creating thumbnail: File:CDel 3.png File:CDel node h.pngError creating thumbnail: File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png File:CDel node.pngError creating thumbnail: File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
File:CDel node h0.pngError creating thumbnail: File:CDel 3.pngFile:CDel node.png
File:CDel label4.pngFile:CDel branch 11.pngFile:CDel split2.pngFile:CDel node.png 		File:CDel node h0.pngError creating thumbnail: File:CDel 3.png
File:CDel label4.pngFile:CDel branch 11.pngFile:CDel split2.png 		File:CDel node h0.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.png
File:CDel label4.pngFile:CDel branch.pngFile:CDel split2.png 		Error creating thumbnail: File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png File:CDel node h1.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:CDel label4.pngFile:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node.png or File:CDel label4.pngFile:CDel branch 01rd.pngFile:CDel split2.pngFile:CDel node.png 		File:CDel node h1.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.png
File:CDel label4.pngFile:CDel branch 10ru.pngFile:CDel split2.png or File:CDel label4.pngFile:CDel branch 01rd.pngFile:CDel split2.png 		File:CDel node h0.pngError creating thumbnail: File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
File:CDel label4.pngFile:CDel branch hh.pngFile:CDel split2.pngFile:CDel node h.png
File:H2-8-3-dual.svg Error creating thumbnail: File:H2-8-3-rectified.svg
File:Uniform tiling 433-t01.png 		File:H2-8-3-trunc-primal.svg
File:Uniform tiling 433-t012.png 		File:H2-8-3-primal.svg
File:Uniform tiling 433-t2.png 		File:H2-8-3-cantellated.svg File:H2-8-3-omnitruncated.svg File:H2-8-3-snub.svg File:Uniform tiling 433-t0.pngFile:Uniform tiling 433-t1.png File:Uniform tiling 433-t02.pngFile:Uniform tiling 433-t12.png File:Uniform tiling 433-snub1.png
File:Uniform tiling 433-snub2.png
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
File:CDel node f1.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node f1.pngError creating thumbnail: File:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngError creating thumbnail: File:CDel node f1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngError creating thumbnail: File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node f1.pngError creating thumbnail: File:CDel node f1.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node fh.pngError creating thumbnail: File:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png File:CDel node fh.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node fh.pngError creating thumbnail: File:CDel node.pngFile:CDel 3.pngFile:CDel node f1.png File:CDel node.pngError creating thumbnail: File:CDel node fh.pngFile:CDel 3.pngFile:CDel node fh.png
File:H2-8-3-primal.svg File:H2-8-3-kis-primal.svg File:H2-8-3-rhombic.svg File:H2-8-3-kis-dual.svg File:H2-8-3-dual.svg File:H2-8-3-deltoidal.svg File:H2-8-3-kisrhombille.svg File:H2-8-3-floret.svg File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t01.png File:Uniform dual tiling 433-snub.png

See also

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Wikimedia Commons has media related to Uniform tiling 3-16-16.

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).
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