Images edit Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tilings edit This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n ) and Coxeter–Dynkin diagram symmetry , being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons .
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 10 forms.
Uniform octagonal/triangular tilings Symmetry: [8,3], (*832) [8,3]+ [1+ ,8,3] [8,3+ ] {8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3} 2 {3,8} tr{8,3} sr{8,3} h{8,3} h2 {8,3} s{3,8} 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
References edit 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. See also edit External links edit Weisstein, Eric W. "Hyperbolic tiling". MathWorld Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld Hyperbolic and Spherical Tiling Gallery KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
snub, trioctagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 8schläfli, symbol, displaystyle, begin, bmatrix, bmatrix, wythoff, symbol, 2coxeter, diagram, orsymmetry, group, dual, order, floret, pe. Snub trioctagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 3 3 3 3 8Schlafli symbol sr 8 3 or s 8 3 displaystyle s begin Bmatrix 8 3 end Bmatrix Wythoff symbol 8 3 2Coxeter diagram or orSymmetry group 8 3 832 Dual Order 8 3 floret pentagonal tilingProperties Vertex transitive ChiralIn geometry the order 3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane There are four triangles one octagon on each vertex It has Schlafli symbol of sr 8 3 Contents 1 Images 2 Related polyhedra and tilings 3 References 4 See also 5 External linksImages editDrawn in chiral pairs with edges missing between black triangles nbsp nbsp Related polyhedra and tilings editThis semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure 3 3 3 3 n and Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp These figures and their duals have n32 rotational symmetry being in the Euclidean plane for n 6 and hyperbolic plane for any higher n The series can be considered to begin with n 2 with one set of faces degenerated into digons n32 symmetry mutations of snub tilings 3 3 3 3 n vteSymmetryn32 Spherical Euclidean Compact hyperbolic Paracomp 232 332 432 532 632 732 832 32Snubfigures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 3 3 3 3 2 3 3 3 3 3 3 3 3 3 4 3 3 3 3 5 3 3 3 3 6 3 3 3 3 7 3 3 3 3 8 3 3 3 3 Gyrofigures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config V3 3 3 3 2 V3 3 3 3 3 V3 3 3 3 4 V3 3 3 3 5 V3 3 3 3 6 V3 3 3 3 7 V3 3 3 3 8 V3 3 3 3 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 10 forms Uniform octagonal triangular tilings vteSymmetry 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Uniform dualsV83 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp References editJohn 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 See also editSnub hexagonal tiling Floret pentagonal tiling Order 3 heptagonal tiling Tilings of regular polygons List of uniform planar tilings Kagome latticeExternal links editWeisstein Eric W Hyperbolic tiling MathWorld Weisstein Eric W Poincare hyperbolic disk MathWorld Hyperbolic and Spherical Tiling Gallery KaleidoTile 3 Educational software to create spherical planar and hyperbolic tilings Hyperbolic Planar Tessellations Don Hatch Retrieved from https en wikipedia org w index php title Snub trioctagonal tiling amp oldid 786602221, wikipedia, wiki , book, books, library,
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