Uniform constructions edit There are 3 lower symmetry uniform construction, one with two colors of apeirogons , one with two colors of squares , and one with two colors of each:
Symmetry (*∞42) (*∞33)+ ,∞,4] = [(∞,4,4)] (*∞∞2)+ ] = [∞,∞] (*∞2∞2)+ ,∞,4,1+ ] Coxeter Schläfli r{∞,4} r{4,∞}1 ⁄2 r{∞,4}1 ⁄2 r{∞,4}1 ⁄4 Coloring Dual
Symmetry edit The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry .
Related polyhedra and tiling edit Paracompact uniform tilings in [∞,4] family {∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4} Dual figures V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43 .∞ V4.8.∞ Alternations [1+ ,∞,4] [∞+ ,4] [∞,1+ ,4] [∞,4+ ] [∞,4,1+ ] [(∞,4,2+ )] [∞,4]+ h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4} Alternation duals V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
Paracompact uniform tilings in [∞,∞] family {∞,∞} t{∞,∞} r{∞,∞} 2t{∞,∞}=t{∞,∞} 2r{∞,∞}={∞,∞} rr{∞,∞} tr{∞,∞} Dual tilings V∞∞ V∞.∞.∞ V(∞.∞)2 V∞.∞.∞ V∞∞ V4.∞.4.∞ V4.4.∞ Alternations [1+ ,∞,∞] [∞+ ,∞] [∞,1+ ,∞] [∞,∞+ ] [∞,∞,1+ ] [(∞,∞,2+ )] [∞,∞]+ h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2 {∞,∞} hrr{∞,∞} sr{∞,∞} Alternation duals V(∞.∞)∞ V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞∞ V(4.∞.4)2 V3.3.∞.3.∞
See also edit
Wikimedia Commons has media related to Uniform tiling 4-i-4-i .
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. External links edit
tetraapeirogonal, tiling, tetraapeirogonal, tilingpoincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 2schläfli, symbol, displaystyle, begin, bmatrix, infty, bmatrix, displaystyle, begin, bmatrix, infty, infty, bmatr. tetraapeirogonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 2Schlafli symbol r 4 or 4 displaystyle begin Bmatrix infty 4 end Bmatrix rr or r displaystyle r begin Bmatrix infty infty end Bmatrix Wythoff symbol 2 4 2Coxeter diagram orSymmetry group 4 42 2 Dual Order 4 infinite rhombille tilingProperties Vertex transitive edge transitiveIn geometry the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schlafli symbol of r 4 Contents 1 Uniform constructions 2 Symmetry 3 Related polyhedra and tiling 4 See also 5 References 6 External linksUniform constructions editThere are 3 lower symmetry uniform construction one with two colors of apeirogons one with two colors of squares and one with two colors of each Symmetry 42 4 33 1 4 4 4 2 4 1 2 2 1 4 1 Coxeter nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Schlafli r 4 r 4 1 2 r 4 1 2 rr r 4 1 4Coloring nbsp nbsp nbsp nbsp Dual nbsp nbsp nbsp nbsp Symmetry editThe dual to this tiling represents the fundamental domains of 2 2 symmetry group The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains creating 2 and 44 symmetry Related polyhedra and tiling edit n42 symmetry mutations of quasiregular tilings 4 n 2 vteSymmetry 4n2 n 4 Spherical Euclidean Compact hyperbolic Paracompact Noncompact 342 3 4 442 4 4 542 5 4 642 6 4 742 7 4 842 8 4 42 4 ni 4 Figures nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 4 3 2 4 4 2 4 5 2 4 6 2 4 7 2 4 8 2 4 2 4 ni 2Paracompact uniform tilings in 4 family vte nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 t 4 r 4 2t 4 t 4 2r 4 4 rr 4 tr 4 Dual figures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 V4 V 4 2 V8 8 V4 V43 V4 8 Alternations 1 4 44 4 2 1 4 2 2 4 4 4 1 2 4 2 2 2 4 42 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 4 s 4 hr 4 s 4 h 4 hrr 4 s 4 nbsp nbsp nbsp nbsp 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 4 V3 3 2 V 4 4 2 V3 3 4 2 V V 44 V3 3 4 3 Paracompact uniform tilings in family vte nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp t r 2t t 2r rr tr Dual tilings nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 V V 2 V V V4 4 V4 4 Alternations 1 2 1 1 2 2 2 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 s hr s h2 hrr sr 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 V V 3 3 V 4 4 V 3 3 V V 4 4 2 V3 3 3 See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 i 4 i List of uniform planar tilings Tilings of regular polygons Uniform tilings in hyperbolic planeReferences 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 External links editWeisstein Eric W Hyperbolic tiling MathWorld Weisstein Eric W Poincare hyperbolic disk MathWorld Retrieved from https en wikipedia org w index php title Tetraapeirogonal tiling amp oldid 1132341540, 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