Constructions edit There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1+ ,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
Two uniform constructions of 4.4.4.6 Name Rhombitetrahexagonal tiling Image Symmetry [6,4]*642 ) [6,1+ ,4] = [∞,3,∞]*3222 ) Schläfli symbol rr{6,4} t0,1,2,3 {∞,3,∞} Coxeter diagram
There are 3 lower symmetry forms seen by including edge-colorings: + ] (4*3) symmetry. + ,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6+ ,4+ ] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.
Lower symmetry constructions + ], (4*3) + ,4], (6*2) + ,4+ ], (32×)
This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of
Symmetry edit The dual tiling, called a deltoidal tetrahexagonal tiling , represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral , with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1+ ,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.
Related polyhedra and tiling edit Uniform tetrahexagonal tilings Symmetry : [6,4], (*642 ) {6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4} Uniform duals V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12 Alternations [1+ ,6,4] [6+ ,4] [6,1+ ,4] [6,4+ ] [6,4,1+ ] [(6,4,2+ )] [6,4]+ h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform tilings in symmetry *3222 64 2 6.4.4.4 2 46
See also edit
Wikimedia Commons has media related to Uniform tiling 4-4-4-6 .
References edit 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 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
rhombitetrahexagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 4schläfli, symbol, displaystyle, begin, bmatrix, bmatrix, wythoff, symbol, 2coxeter, diagramsymmetry, group, dual, deltoidal, tetrahex. Rhombitetrahexagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 4 6 4Schlafli symbol rr 6 4 or r 64 displaystyle r begin Bmatrix 6 4 end Bmatrix Wythoff symbol 4 6 2Coxeter diagramSymmetry group 6 4 642 Dual Deltoidal tetrahexagonal tilingProperties Vertex transitiveIn geometry the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane It has Schlafli symbol of rr 6 4 It can be seen as constructed as a rectified tetrahexagonal tiling r 6 4 as well as an expanded order 4 hexagonal tiling or expanded order 6 square tiling Contents 1 Constructions 2 Symmetry 3 Related polyhedra and tiling 4 See also 5 References 6 External linksConstructions editThere are two uniform constructions of this tiling one from 6 4 or 642 symmetry and secondly removing the mirror middle 6 1 4 gives a rectangular fundamental domain 3 3222 Two uniform constructions of 4 4 4 6 Name Rhombitetrahexagonal tilingImage nbsp nbsp Symmetry 6 4 642 nbsp nbsp nbsp nbsp nbsp 6 1 4 3 3222 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Schlafli symbol rr 6 4 t0 1 2 3 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp There are 3 lower symmetry forms seen by including edge colorings nbsp nbsp nbsp nbsp nbsp sees the hexagons as truncated triangles with two color edges with 6 4 4 3 symmetry nbsp nbsp nbsp nbsp nbsp sees the yellow squares as rectangles with two color edges with 6 4 6 2 symmetry A final quarter symmetry combines these colorings with 6 4 32 symmetry with 2 and 3 fold gyration points and glide reflections Lower symmetry constructions nbsp nbsp nbsp nbsp nbsp nbsp 6 4 632 nbsp nbsp nbsp nbsp nbsp nbsp 6 4 4 3 nbsp nbsp nbsp nbsp nbsp nbsp 6 4 6 2 nbsp nbsp 6 4 32 This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3 space with a prismatic honeycomb construction of nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry editThe dual tiling called a deltoidal tetrahexagonal tiling represents the fundamental domains of the 3222 orbifold shown here from three different centers Its fundamental domain is a Lambert quadrilateral with 3 right angles This symmetry can be seen from a 6 4 642 triangular symmetry with one mirror removed constructed as 6 1 4 3222 Removing half of the blue mirrors doubles the domain again into 3322 symmetry nbsp nbsp nbsp nbsp Related polyhedra and tiling edit n42 symmetry mutation of expanded tilings n 4 4 4 vteSymmetry n 4 n42 Spherical Euclidean Compact hyperbolic Paracomp 342 3 4 442 4 4 542 5 4 642 6 4 742 7 4 842 8 4 42 4 Expandedfigures nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 3 4 4 4 4 4 4 4 5 4 4 4 6 4 4 4 7 4 4 4 8 4 4 4 4 4 4Rhombicfiguresconfig nbsp V3 4 4 4 nbsp V4 4 4 4 nbsp V5 4 4 4 nbsp V6 4 4 4 nbsp V7 4 4 4 nbsp V8 4 4 4 nbsp V 4 4 4Uniform tetrahexagonal tilings vteSymmetry 6 4 642 with 6 6 662 4 3 3 443 3 3222 index 2 subsymmetries And 3 3 3232 index 4 subsymmetry 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 6 4 t 6 4 r 6 4 t 4 6 4 6 rr 6 4 tr 6 4 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 V64 V4 12 12 V 4 6 2 V6 8 8 V46 V4 4 4 6 V4 8 12Alternations 1 6 4 443 6 4 6 2 6 1 4 3222 6 4 4 3 6 4 1 662 6 4 2 2 32 6 4 642 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 h 6 4 s 6 4 hr 6 4 s 4 6 h 4 6 hrr 6 4 sr 6 4 Uniform tilings in symmetry 3222 vte nbsp nbsp nbsp 64 nbsp nbsp nbsp nbsp 6 6 4 4 nbsp nbsp nbsp nbsp 3 4 4 2 nbsp nbsp nbsp nbsp 4 3 4 3 3 3 nbsp nbsp nbsp nbsp 6 6 4 4 nbsp nbsp nbsp nbsp 6 4 4 4 nbsp nbsp nbsp nbsp 3 4 4 4 4 nbsp nbsp nbsp nbsp 3 4 4 2 nbsp nbsp nbsp nbsp 3 4 4 4 4 nbsp nbsp nbsp nbsp 46 nbsp See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 4 4 6 Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopesReferences editJohn 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 Retrieved from https en wikipedia org w index php title Rhombitetrahexagonal tiling amp oldid 1189602048 Symmetry, wikipedia, wiki , book, books, library,
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