Uniform constructions edit There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope . Removing the mirror between the order 2 and 4 points, [8,8,1+ ], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4* ], leaves remaining mirrors *4444 symmetry.
Four uniform constructions of 8.8.8.8 Uniform Symmetry [8,4] [8,8] [(8,4,8)] = [8,8,1+ ]
[1+ ,8,8,1+ ] Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}1 ⁄2 r{8,4}1 ⁄8 1 ⁄4 Coxeter
Symmetry edit This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28 ) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8* ,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry . Adding 4 bisecting mirrors through the edge defines *4222 symmetry . Adding all 8 bisectors leads to full *842 symmetry .
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling .
Related polyhedra and tiling edit This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling , with Schläfli symbol {8,n}, and Coxeter diagram
*n 42 symmetry mutation of regular tilings: {n ,4} Spherical Euclidean Hyperbolic tilings 24 34 44 54 64 74 84 ...∞4
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron , with Schläfli symbol {n,4}, and Coxeter diagram
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 octaoctagonal tilings Symmetry: [8,8], (*882) {8,8} t{8,8} r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8} Uniform duals V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16 Alternations [1+ ,8,8] [8+ ,8] [8,1+ ,8] [8,8+ ] [8,8,1+ ] [(8,8,2+ )] [8,8]+ h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8} Alternation duals V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
See also edit
Wikimedia Commons has media related to Order-4 octagonal 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, octagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, regular, tilingvertex, configuration, 84schläfli, symbol, wythoff, symbol, 2coxeter, diagram, orsymmetry, group, dual, order, square, tilingproperties, vertex, transitive, edge. Order 4 octagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic regular tilingVertex configuration 84Schlafli symbol 8 4 r 8 8 Wythoff symbol 4 8 2Coxeter diagram orSymmetry group 8 4 842 8 8 882 Dual Order 8 square tilingProperties Vertex transitive edge transitive face transitiveIn geometry the order 4 octagonal tiling is a regular tiling of the hyperbolic plane It has Schlafli symbol of 8 4 Its checkerboard coloring can be called a octaoctagonal tiling and Schlafli symbol of r 8 8 Contents 1 Uniform constructions 2 Symmetry 3 Related polyhedra and tiling 4 See also 5 References 6 External linksUniform constructions editThere are four uniform constructions of this tiling three of them as constructed by mirror removal from the 8 8 kaleidoscope Removing the mirror between the order 2 and 4 points 8 8 1 gives 8 8 4 884 symmetry Removing two mirrors as 8 4 leaves remaining mirrors 4444 symmetry Four uniform constructions of 8 8 8 8 UniformColoring nbsp nbsp nbsp nbsp Symmetry 8 4 842 nbsp nbsp nbsp nbsp nbsp 8 8 882 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 8 4 8 8 8 1 884 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 8 8 1 4444 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symbol 8 4 r 8 8 r 8 4 8 r 8 8 1 2 r 8 4 1 8 r 8 8 1 4Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry editThis tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon This symmetry by orbifold notation is called 22222222 or 28 with 8 order 2 mirror intersections In Coxeter notation can be represented as 8 4 removing two of three mirrors passing through the octagon center in the 8 4 symmetry Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral 4422 symmetry Adding 4 bisecting mirrors through the vertices defines 444 symmetry Adding 4 bisecting mirrors through the edge defines 4222 symmetry Adding all 8 bisectors leads to full 842 symmetry nbsp 444 nbsp 4222 nbsp 832The kaleidoscopic domains can be seen as bicolored octagonal tiling representing mirror images of the fundamental domain This coloring represents the uniform tiling r 8 8 a quasiregular tiling and it can be called a octaoctagonal tiling nbsp nbsp Related polyhedra and tiling editThis tiling is topologically related as a part of sequence of regular tilings with octagonal faces starting with the octagonal tiling with Schlafli symbol 8 n and Coxeter diagram nbsp nbsp nbsp nbsp nbsp progressing to infinity n42 symmetry mutation of regular tilings n 4 vteSpherical Euclidean Hyperbolic tilings nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 24 34 44 54 64 74 84 4Regular tilings n 8 vteSpherical Hyperbolic tilings nbsp 2 8 nbsp nbsp nbsp nbsp nbsp nbsp 3 8 nbsp nbsp nbsp nbsp nbsp nbsp 4 8 nbsp nbsp nbsp nbsp nbsp nbsp 5 8 nbsp nbsp nbsp nbsp nbsp nbsp 6 8 nbsp nbsp nbsp nbsp nbsp nbsp 7 8 nbsp nbsp nbsp nbsp nbsp nbsp 8 8 nbsp nbsp nbsp nbsp nbsp nbsp 8 nbsp nbsp nbsp nbsp nbsp This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex starting with the octahedron with Schlafli symbol n 4 and Coxeter diagram nbsp nbsp nbsp nbsp nbsp with n progressing to infinity nbsp 3 4 nbsp nbsp nbsp nbsp nbsp nbsp 4 4 nbsp nbsp nbsp nbsp nbsp nbsp 5 4 nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp 7 4 nbsp nbsp nbsp nbsp nbsp nbsp 8 4 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 16Alternations 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 8Uniform octaoctagonal tilings vteSymmetry 8 8 882 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 8 t 8 8 r 8 8 2t 8 8 t 8 8 2r 8 8 8 8 rr 8 8 tr 8 8 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 V88 V8 16 16 V8 8 8 8 V8 16 16 V88 V4 8 4 8 V4 16 16Alternations 1 8 8 884 8 8 8 4 8 1 8 4242 8 8 8 4 8 8 1 884 8 8 2 2 44 8 8 882 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 8 s 8 8 hr 8 8 s 8 8 h 8 8 hrr 8 8 sr 8 8 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 V 4 8 8 V3 4 3 8 3 8 V 4 4 4 V3 4 3 8 3 8 V 4 8 8 V46 V3 3 8 3 8See also edit nbsp Wikimedia Commons has media related to Order 4 octagonal tiling 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 Order 4 octagonal tiling amp oldid 1189601605, wikipedia, wiki , book, books, library,
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