In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball,[1] is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.
This tiling is called a hyperbolic soccerball (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space.
The dual tiling is called a heptakis heptagonal tiling, named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point.
^HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
"Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN0-486-40919-8. LCCN 99035678.
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Geometric explorations on the hyperbolic football by Frank Sottile
April 11, 2024
truncated, order, triangular, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 6schläfli, symbol, wythoff, symbol, 3coxeter, diagramsymmetry, group, dual, heptakis, heptagonal, tilingproperties, vertex, tr. Truncated order 7 triangular tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 7 6 6Schlafli symbol t 3 7 Wythoff symbol 2 7 3Coxeter diagramSymmetry group 7 3 732 Dual Heptakis heptagonal tilingProperties Vertex transitiveIn geometry the order 7 truncated triangular tiling sometimes called the hyperbolic soccerball 1 is a semiregular tiling of the hyperbolic plane There are two hexagons and one heptagon on each vertex forming a pattern similar to a conventional soccer ball truncated icosahedron with heptagons in place of pentagons It has Schlafli symbol of t 3 7 Contents 1 Hyperbolic soccerball football 2 Dual tiling 3 Related tilings 4 In popular culture 5 See also 6 References 7 External linksHyperbolic soccerball football editThis tiling is called a hyperbolic soccerball football for its similarity to the truncated icosahedron pattern used on soccer balls Small portions of it as a hyperbolic surface can be constructed in 3 space nbsp A truncated icosahedronas a polyhedron and a ball nbsp The Euclidean hexagonal tilingcolored as truncatedtriangular tiling nbsp A paper constructionof a hyperbolic soccerballDual tiling editThe dual tiling is called a heptakis heptagonal tiling named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point nbsp Related tilings editThis hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations n 6 6 and n 3 Coxeter group symmetry n32 symmetry mutation of truncated tilings n 6 6 vteSym n42 n 3 Spherical Euclid Compact Parac 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 Truncatedfigures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 2 6 6 3 6 6 4 6 6 5 6 6 6 6 6 7 6 6 8 6 6 6 6 12i 6 6 9i 6 6 6i 6 6n kisfigures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config V2 6 6 V3 6 6 V4 6 6 V5 6 6 V6 6 6 V7 6 6 V8 6 6 V 6 6 V12i 6 6 V9i 6 6 V6i 6 6From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal 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 heptagonal triangular tilings vteSymmetry 7 3 732 7 3 732 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 7 3 t 7 3 r 7 3 t 3 7 3 7 rr 7 3 tr 7 3 sr 7 3 Uniform duals 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 V73 V3 14 14 V3 7 3 7 V6 6 7 V37 V3 4 7 4 V4 6 14 V3 3 3 3 7In popular culture editThis tiling features prominently in HyperRogue See also edit nbsp Wikimedia Commons has media related to Uniform tiling 6 6 7 Triangular tiling Order 3 heptagonal tiling Order 7 triangular tiling Tilings of regular polygons List of uniform tilingsReferences edit HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL John H Conway Heidi Burgiel Chaim Goodman Strauss 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 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 Geometric explorations on the hyperbolic football by Frank Sottile Retrieved from https en wikipedia org w index php title Truncated order 7 triangular tiling amp oldid 1189601564, wikipedia, wiki, book, books, library,
