Symmetry edit This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as [1+ ,8,8,1+ ], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] symmetry . The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry .
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4[3] },
Related polyhedra and tiling edit This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n ).
Uniform octagonal/square tilings [8,4], (*842) {8,4} t{8,4} r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4} Uniform duals V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16 Alternations [1+ ,8,4] [8+ ,4] [8,1+ ,4] [8,4+ ] [8,4,1+ ] [(8,4,2+ )] [8,4]+ h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4} Alternation duals V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform (4,4,4) tilings Symmetry: [(4,4,4)], (*444) [(4,4,4)]+ [(1+ ,4,4,4)] [(4+ ,4,4)] t0 (4,4,4) t0,1 (4,4,4) 2 {8,4} t1 (4,4,4) 1 /2 t1,2 (4,4,4) 2 {8,4} t2 (4,4,4) t0,2 (4,4,4) 1 /2 t0,1,2 (4,4,4) 1 /2 s(4,4,4) 1 /2 h(4,4,4) 1 /2 hr(4,4,4) 1 /2 Uniform duals V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3
See also edit
Wikimedia Commons has media related to Order-8 square tiling .
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
order, square, tiling, poincaré, disk, model, hyperbolic, plane, type, hyperbolic, regular, tiling, vertex, configuration, schläfli, symbol, wythoff, symbol, coxeter, diagram, symmetry, group, dual, order, octagonal, tiling, properties, vertex, transitive, edg. Order 8 square tiling Poincare disk model of the hyperbolic plane Type Hyperbolic regular tiling Vertex configuration 48 Schlafli symbol 4 8 Wythoff symbol 8 4 2 Coxeter diagram Symmetry group 8 4 842 Dual Order 4 octagonal tiling Properties Vertex transitive edge transitive face transitive In geometry the order 8 square tiling is a regular tiling of the hyperbolic plane It has Schlafli symbol of 4 8 Contents 1 Symmetry 2 Related polyhedra and tiling 3 See also 4 References 5 External linksSymmetry editThis tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square with eight squares around every vertex This symmetry by orbifold notation is called 4444 with 4 order 4 mirror intersections In Coxeter notation can be represented as 1 8 8 1 4444 orbifold removing two of three mirrors passing through the square center in the 8 8 symmetry The 4444 symmetry can be doubled by bisecting the fundamental domain square by a mirror creating 884 symmetry This bicolored square tiling shows the even odd reflective fundamental square domains of this symmetry This bicolored tiling has a wythoff construction 4 4 4 or 4 3 nbsp nbsp nbsp nbsp nbsp nbsp Related polyhedra and tiling editThis tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure 4n n42 symmetry mutation of regular tilings 4 n vte Spherical Euclidean Compact hyperbolic Paracompact nbsp 4 3 nbsp nbsp nbsp nbsp nbsp nbsp 4 4 nbsp nbsp nbsp nbsp nbsp nbsp 4 5 nbsp nbsp nbsp nbsp nbsp nbsp 4 6 nbsp nbsp nbsp nbsp nbsp nbsp 4 7 nbsp nbsp nbsp nbsp nbsp nbsp 4 8 nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp Uniform octagonal square tilings vte 8 4 842 with 8 8 882 4 4 4 444 4 4222 index 2 subsymmetries And 4 4 4242 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 nbsp nbsp nbsp nbsp nbsp 8 4 t 8 4 r 8 4 2t 8 4 t 4 8 2r 8 4 4 8 rr 8 4 tr 8 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 V84 V4 16 16 V 4 8 2 V8 8 8 V48 V4 4 4 8 V4 8 16 Alternations 1 8 4 444 8 4 8 2 8 1 4 4222 8 4 4 4 8 4 1 882 8 4 2 2 42 8 4 842 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 8 4 s 8 4 hr 8 4 s 4 8 h 4 8 hrr 8 4 sr 8 4 Alternation 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 V 4 4 4 V3 3 8 2 V 4 4 4 2 V 3 4 3 V88 V4 44 V3 3 4 3 8 Uniform 4 4 4 tilings vte Symmetry 4 4 4 444 4 4 4 444 1 4 4 4 4242 4 4 4 4 22 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp t0 4 4 4 h 8 4 t0 1 4 4 4 h2 8 4 t1 4 4 4 4 8 1 2 t1 2 4 4 4 h2 8 4 t2 4 4 4 h 8 4 t0 2 4 4 4 r 4 8 1 2 t0 1 2 4 4 4 t 4 8 1 2 s 4 4 4 s 4 8 1 2 h 4 4 4 h 4 8 1 2 hr 4 4 4 hr 4 8 1 2 Uniform duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 4 4 4 V4 8 4 8 V 4 4 4 V4 8 4 8 V 4 4 4 V4 8 4 8 V8 8 8 V3 4 3 4 3 4 V88 V 4 4 3See also edit nbsp Wikimedia Commons has media related to Order 8 square tiling Square tiling Uniform tilings in hyperbolic plane 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 Order 8 square tiling amp oldid 1196860588, wikipedia, wiki , book, books, library,
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