Dual tiling edit The dual tiling is called an order-4-6 kisrhombille tiling , made as a complete bisection of the order-4 hexagonal tiling , here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,4] (*642) symmetry.
Related polyhedra and tilings edit From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,4] symmetry, and 7 with subsymmetry.
Uniform tetrahexagonal tilings Symmetry : [6,4], (*642 ) {6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4} Uniform duals V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12 Alternations [1+ ,6,4] [6+ ,4] [6,1+ ,4] [6,4+ ] [6,4,1+ ] [(6,4,2+ )] [6,4]+ h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Symmetry edit Truncated tetrahexagonal tiling with mirror lines in green, red, and blue: Symmetry diagrams for small index subgroups of [6,4], shown in a hexagonal translational cell within a {6,6} tiling , with a fundamental domain in yellow. The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry. From [6,4] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. 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 unique mirrors are colored red, green, and blue, and alternately colored triangles show the location of gyration points. The [6+ ,4+ ], (32×) subgroup has narrow lines representing glide reflections. The subgroup index -8 group, [1+ ,6,1+ ,4,1+ ] (3232) is the commutator subgroup of [6,4].
Larger subgroup constructed as [6,4*], removing the gyration points of [6,4+ ], (3*22), index 6 becomes (*3333 ), and [6*,4], removing the gyration points of [6+ ,4], (2*33), index 12 as (*222222 ). Finally their direct subgroups [6,4*]+ , [6*,4]+ , subgroup indices 12 and 24 respectively, can be given in orbifold notation as (3333) and (222222).
Small index subgroups of [6,4] Index 1 2 4 Diagram Coxeter [6,4] [1+ ,6,4] [6,4,1+ ] [6,1+ ,4] [1+ ,6,4,1+ ] [6+ ,4+ ] Generators {0 ,1 ,2 } {1 ,010 ,2 } {0 ,1 ,212 } {0 ,101 ,2 ,121 } {1 ,010 ,212 ,20102 } {012,021} Orbifold *642 *443 *662 *3222 *3232 32× Semidirect subgroups Diagram Coxeter [6,4+ ] [6+ ,4] [(6,4,2+ )] [6,1+ ,4,1+ ] [1+ ,6,1+ ,4] Generators {0 ,12} {01,2 } {1 ,02} {0 ,101 ,1212} {0101,2 ,121 } Orbifold 4*3 6*2 2*32 2*33 3*22 Direct subgroups Index 2 4 8 Diagram Coxeter [6,4]+ [6,4+ ]+ [6+ ,4]+ [(6,4,2+ )]+ [6+ ,4+ ]+ = [1+ ,6,1+ ,4,1+ ] Generators {01,12} {(01)2 ,12} {01,(12)2 } {02,(01)2 ,(12)2 } {(01)2 ,(12)2 ,2(01)2 2} Orbifold 642 443 662 3222 3232 Radical subgroups Index 8 12 16 24 Diagram Coxeter [6,4*] [6*,4] [6,4*]+ [6*,4]+ Orbifold *3333 *222222 3333 222222
See also edit
Wikimedia Commons has media related to Uniform tiling 4-8-12 .
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
truncated, tetrahexagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 12schläfli, symbol, displaystyle, begin, bmatrix, bmatrix, wythoff, symbol, coxeter, diagram, orsymmetry, group, dual, order, kis. Truncated tetrahexagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 8 12Schlafli symbol tr 6 4 or t 6 4 displaystyle t begin Bmatrix 6 4 end Bmatrix Wythoff symbol 2 6 4 Coxeter diagram orSymmetry group 6 4 642 Dual Order 4 6 kisrhombille tilingProperties Vertex transitiveIn geometry the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane There are one square one octagon and one dodecagon on each vertex It has Schlafli symbol of tr 6 4 Contents 1 Dual tiling 2 Related polyhedra and tilings 3 Symmetry 4 See also 5 References 6 External linksDual tiling edit nbsp nbsp The dual tiling is called an order 4 6 kisrhombille tiling made as a complete bisection of the order 4 hexagonal tiling here with triangles shown in alternating colors This tiling represents the fundamental triangular domains of 6 4 642 symmetry Related polyhedra and tilings edit 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 From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order 4 hexagonal tiling Drawing the tiles colored as red on the original faces yellow at the original vertices and blue along the original edges there are 7 forms with full 6 4 symmetry and 7 with subsymmetry Uniform tetrahexagonal tilings vteSymmetry 6 4 642 with 6 6 662 4 3 3 443 3 3222 index 2 subsymmetries And 3 3 3232 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 6 4 t 6 4 r 6 4 t 4 6 4 6 rr 6 4 tr 6 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 V64 V4 12 12 V 4 6 2 V6 8 8 V46 V4 4 4 6 V4 8 12Alternations 1 6 4 443 6 4 6 2 6 1 4 3222 6 4 4 3 6 4 1 662 6 4 2 2 32 6 4 642 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 6 4 s 6 4 hr 6 4 s 4 6 h 4 6 hrr 6 4 sr 6 4 Symmetry edit nbsp Truncated tetrahexagonal tiling with mirror lines in green red and blue nbsp nbsp nbsp nbsp nbsp nbsp Symmetry diagrams for small index subgroups of 6 4 shown in a hexagonal translational cell within a 6 6 tiling with a fundamental domain in yellow The dual of the tiling represents the fundamental domains of 642 orbifold symmetry From 6 4 symmetry there are 15 small index subgroup by mirror removal and alternation operators 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 unique mirrors are colored red green and blue and alternately colored triangles show the location of gyration points The 6 4 32 subgroup has narrow lines representing glide reflections The subgroup index 8 group 1 6 1 4 1 3232 is the commutator subgroup of 6 4 Larger subgroup constructed as 6 4 removing the gyration points of 6 4 3 22 index 6 becomes 3333 and 6 4 removing the gyration points of 6 4 2 33 index 12 as 222222 Finally their direct subgroups 6 4 6 4 subgroup indices 12 and 24 respectively can be given in orbifold notation as 3333 and 222222 Small index subgroups of 6 4 Index 1 2 4Diagram nbsp nbsp nbsp nbsp nbsp nbsp Coxeter 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 6 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp Generators 0 1 2 1 010 2 0 1 212 0 101 2 121 1 010 212 20102 012 021 Orbifold 642 443 662 3222 3232 32 Semidirect subgroupsDiagram nbsp nbsp nbsp nbsp nbsp Coxeter 6 4 nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp 6 4 2 nbsp nbsp nbsp nbsp 6 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 1 6 1 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Generators 0 12 01 2 1 02 0 101 1212 0101 2 121 Orbifold 4 3 6 2 2 32 2 33 3 22Direct subgroupsIndex 2 4 8Diagram nbsp nbsp nbsp nbsp nbsp Coxeter 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 1 6 1 4 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Generators 01 12 01 2 12 01 12 2 02 01 2 12 2 01 2 12 2 2 01 22 Orbifold 642 443 662 3222 3232Radical subgroupsIndex 8 12 16 24Diagram nbsp nbsp nbsp nbsp Coxeter 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 nbsp nbsp nbsp nbsp nbsp nbsp Orbifold 3333 222222 3333 222222See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 8 12 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 KaleidoTile 3 Educational software to create spherical planar and hyperbolic tilings Hyperbolic Planar Tessellations Don Hatch 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