Symmetry edit Truncated triapeirogonal tiling with mirrors The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+ , (∞∞3), and semidirect subgroup [(∞,∞,3+ )], (3*∞).[1]
An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).
Small index subgroups of [∞,3], (*∞32) Index 1 2 3 4 6 8 12 24 Diagrams Coxeter orbifold ) [∞,3] [1+ ,∞,3] *∞33 ) [∞,3+ ] [∞,∞]*∞∞2 ) [(∞,∞,3)] [∞,3*] *∞3 ) [∞,1+ ,∞]2 ) [(∞,1+ ,∞,3)]2 ) [1+ ,∞,∞,1+ ](*∞4 ) [(∞,∞,3*)]6 ) Direct subgroups Index 2 4 6 8 12 16 24 48 Diagrams Coxeter [∞,3]+ [∞,3+ ]+ [∞,∞]+ [(∞,∞,3)]+ [∞,3*]+ 3 ) [∞,1+ ,∞]+ 2 [(∞,1+ ,∞,3)]+ 2 [1+ ,∞,∞,1+ ]+ 4 ) [(∞,∞,3*)]+ 6 )
Related polyhedra and tiling edit Paracompact uniform tilings in [∞,3] family Symmetry: [∞,3], (*∞32) [∞,3]+ [1+ ,∞,3] [∞,3+ ] {∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2 {∞,3} s{3,∞} Uniform duals V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3∞ V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons ), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling .
*n 32 symmetry mutation of omnitruncated tilings: 4.6.2n Sym.*n 32 [n ,3] Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic *232 *332 *432 *532 *632 *732 *832 *∞32 Figures Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i Duals Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
See also edit
Wikimedia Commons has media related to Uniform tiling 4-6-i .
References edit ^ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups , Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [1] 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
truncated, triapeirogonal, tiling, poincaré, disk, model, hyperbolic, plane, type, hyperbolic, uniform, tiling, vertex, configuration, schläfli, symbol, displaystyle, begin, bmatrix, infty, bmatrix, wythoff, symbol, coxeter, diagram, symmetry, group, dual, ord. Truncated triapeirogonal tiling Poincare disk model of the hyperbolic plane Type Hyperbolic uniform tiling Vertex configuration 4 6 Schlafli symbol tr 3 or t 3 displaystyle t begin Bmatrix infty 3 end Bmatrix Wythoff symbol 2 3 Coxeter diagram or Symmetry group 3 32 Dual Order 3 infinite kisrhombille Properties Vertex transitive In geometry the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schlafli symbol of tr 3 Contents 1 Symmetry 2 Related polyhedra and tiling 3 See also 4 References 5 External linksSymmetry edit nbsp Truncated triapeirogonal tiling with mirrors The dual of this tiling represents the fundamental domains of 3 32 symmetry There are 3 small index subgroup constructed from 3 by mirror removal and alternation In these images fundamental domains are alternately colored black and white and mirrors exist on the boundaries between colors A special index 4 reflective subgroup is 3 3 and its direct subgroup 3 3 and semidirect subgroup 3 3 1 Given 3 with generating mirrors 0 1 2 then its index 4 subgroup has generators 0 121 212 An index 6 subgroup constructed as 3 becomes Small index subgroups of 3 32 Index 1 2 3 4 6 8 12 24 Diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coxeter orbifold 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 32 1 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 33 3 nbsp nbsp nbsp nbsp nbsp 3 2 3 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 1 2 2 1 3 3 2 1 1 4 3 6 Direct subgroups Index 2 4 6 8 12 16 24 48 Diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coxeter orbifold 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 32 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 33 2 3 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 1 2 2 1 3 3 2 1 1 4 3 6 Related polyhedra and tiling editParacompact uniform tilings in 3 family vte Symmetry 3 32 3 32 1 3 33 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 t 3 r 3 t 3 3 rr 3 tr 3 sr 3 h 3 h2 3 s 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V 3 V3 V 3 2 V6 6 V3 V4 3 4 V4 6 V3 3 3 3 V 3 3 V3 3 3 3 3 This tiling can be considered a member of a sequence of uniform patterns with vertex figure 4 6 2p and Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp For p lt 6 the members of the sequence are omnitruncated polyhedra zonohedrons shown below as spherical tilings For p gt 6 they are tilings of the hyperbolic plane starting with the truncated triheptagonal tiling n32 symmetry mutation of omnitruncated tilings 4 6 2n vte Sym n32 n 3 Spherical Euclid Compact hyperb Paraco 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 3i 3 Figures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 4 6 4 4 6 6 4 6 8 4 6 10 4 6 12 4 6 14 4 6 16 4 6 4 6 24i 4 6 18i 4 6 12i 4 6 6i Duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config V4 6 4 V4 6 6 V4 6 8 V4 6 10 V4 6 12 V4 6 14 V4 6 16 V4 6 V4 6 24i V4 6 18i V4 6 12i V4 6 6iSee also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 6 i List of uniform planar tilings Tilings of regular polygons Uniform tilings in hyperbolic planeReferences edit Norman W Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups Can J Math Vol 51 6 1999 pp 1307 1336 1 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 Retrieved from https en wikipedia org w index php title Truncated triapeirogonal tiling amp oldid 1189601991 Symmetry, wikipedia, wiki , book, books, library,
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