Related polyhedra and tilings 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.∞
Symmetry edit The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index -8 group, [1+ ,∞,1+ ,4,1+ ] (∞2∞2) is the commutator subgroup of [∞,4].
A larger subgroup is constructed as [∞,4*], index 8, as [∞,4+ ], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞4 ), and another [∞*,4], index ∞ as [∞+ ,4], (∞*2) with gyration points removed as (*2∞ ). And their direct subgroups [∞,4*]+ , [∞*,4]+ , subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2∞ ).
Small index subgroups of [∞,4], (*∞42) Index 1 2 4 Diagram Coxeter [∞,4] [1+ ,∞,4] [∞,4,1+ ] [∞,1+ ,4] [1+ ,∞,4,1+ ] [∞+ ,4+ ] Orbifold *∞42 *∞44 *∞∞2 *∞222 *∞2∞2 ∞2× Semidirect subgroups Diagram Coxeter [∞,4+ ] [∞+ ,4] [(∞,4,2+ )] [1+ ,∞,1+ ,4] [∞,1+ ,4,1+ ] Orbifold 4*∞ ∞*2 2*∞2 ∞*22 2*∞∞ Direct subgroups Index 2 4 8 Diagram Coxeter [∞,4]+ [∞,4+ ]+ [∞+ ,4]+ [∞,1+ ,4]+ [∞+ ,4+ ]+ = [1+ ,∞,1+ ,4,1+ ] Orbifold ∞42 ∞44 ∞∞2 ∞222 ∞2∞2 Radical subgroups Index 8 ∞ 16 ∞ Diagram Coxeter [∞,4*] [∞*,4] [∞,4*]+ [∞*,4]+ Orbifold *∞∞∞∞ *2∞ ∞∞∞∞ 2∞
See also edit
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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
truncated, tetraapeirogonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, schläfli, symbol, displaystyle, begin, bmatrix, infty, bmatrix, wythoff, symbol, coxeter, diagram, orsymmetry, group, dual, ord. Truncated tetraapeirogonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 8 Schlafli symbol tr 4 or t 4 displaystyle t begin Bmatrix infty 4 end Bmatrix Wythoff symbol 2 4 Coxeter diagram orSymmetry group 4 42 Dual Order 4 infinite kisrhombilleProperties Vertex transitiveIn geometry the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane There are one square one octagon and one apeirogon on each vertex It has Schlafli symbol of tr 4 Contents 1 Related polyhedra and tilings 2 Symmetry 3 See also 4 References 5 External linksRelated polyhedra and tilings editParacompact 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 n42 symmetry mutation of omnitruncated tilings 4 8 2n vteSymmetry n42 n 4 Spherical Euclidean Compact hyperbolic Paracomp 242 2 4 342 3 4 442 4 4 542 5 4 642 6 4 742 7 4 842 8 4 42 4 Omnitruncatedfigure nbsp 4 8 4 nbsp 4 8 6 nbsp 4 8 8 nbsp 4 8 10 nbsp 4 8 12 nbsp 4 8 14 nbsp 4 8 16 nbsp 4 8 Omnitruncatedduals nbsp V4 8 4 nbsp V4 8 6 nbsp V4 8 8 nbsp V4 8 10 nbsp V4 8 12 nbsp V4 8 14 nbsp V4 8 16 nbsp V4 8 nn2 symmetry mutations of omnitruncated tilings 4 2n 2n vteSymmetry nn2 n n Spherical Euclidean Compact hyperbolic Paracomp 222 2 2 332 3 3 442 4 4 552 5 5 662 6 6 772 7 7 882 8 8 2 Figure nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 4 4 4 4 6 6 4 8 8 4 10 10 4 12 12 4 14 14 4 16 16 4 Dual nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config V4 4 4 V4 6 6 V4 8 8 V4 10 10 V4 12 12 V4 14 14 V4 16 16 V4 Symmetry editThe dual of this tiling represents the fundamental domains of 4 42 symmetry There are 15 small index subgroups constructed from 4 by mirror removal and alternation Mirrors can be removed if its branch orders are all even and cuts neighboring branch orders in half Removing two mirrors leaves a half order gyration point where the removed mirrors met In these images fundamental domains are alternately colored black and white and mirrors exist on the boundaries between colors The subgroup index 8 group 1 1 4 1 2 2 is the commutator subgroup of 4 A larger subgroup is constructed as 4 index 8 as 4 4 with gyration points removed becomes or 4 and another 4 index as 4 2 with gyration points removed as 2 And their direct subgroups 4 4 subgroup indices 16 and respectively can be given in orbifold notation as and 2 Small index subgroups of 4 42 Index 1 2 4Diagram nbsp nbsp nbsp nbsp nbsp nbsp Coxeter 4 nbsp nbsp nbsp nbsp nbsp 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp Orbifold 42 44 2 222 2 2 2 Semidirect subgroupsDiagram nbsp nbsp nbsp nbsp nbsp Coxeter 4 nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp 4 2 nbsp nbsp nbsp nbsp 1 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Orbifold 4 2 2 2 22 2 Direct subgroupsIndex 2 4 8Diagram nbsp nbsp nbsp nbsp nbsp Coxeter 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 1 1 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Orbifold 42 44 2 222 2 2Radical subgroupsIndex 8 16 Diagram nbsp nbsp nbsp nbsp Coxeter 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 nbsp nbsp nbsp nbsp nbsp nbsp Orbifold 2 2 See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 8 i Tilings of regular polygons List of uniform planar tilingsReferences 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 Retrieved from https en wikipedia org w index php title Truncated tetraapeirogonal tiling amp oldid 1189602260 Symmetry, wikipedia, wiki , book, books, library,
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