[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-tetrahexagonal-tiling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-tetrahexagonal-tiling-wikipedia\/","headline":"Truncated tetrahexagonal tiling – Wikipedia","name":"Truncated tetrahexagonal tiling – Wikipedia","description":"From Wikipedia, the free encyclopedia In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There","datePublished":"2015-10-08","dateModified":"2015-10-08","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/52\/H2checkers_246.png\/160px-H2checkers_246.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/52\/H2checkers_246.png\/160px-H2checkers_246.png","height":"160","width":"160"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-tetrahexagonal-tiling-wikipedia\/","wordCount":12950,"articleBody":"From Wikipedia, the free encyclopediaIn 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 Schl\u00e4fli symbol of tr{6,4}.Table of ContentsDual tiling[edit]Related polyhedra and tilings[edit]Symmetry[edit]See also[edit]References[edit]External links[edit]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)(with [6,6] (*662), [(4,3,3)] (*443) , [\u221e,3,\u221e] (*3222) index 2 subsymmetries)(And [(\u221e,3,\u221e,3)] (*3232) index 4 subsymmetry)= = = = = = = = = = = = {6,4}t{6,4}r{6,4}t{4,6}{4,6}rr{6,4}tr{6,4}Uniform dualsV64V4.12.12V(4.6)2V6.8.8V46V4.4.4.6V4.8.12Alternations[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)= = = = = = h{6,4}s{6,4}hr{6,4}s{4,6}h{4,6}hrr{6,4}sr{6,4}Symmetry[edit]    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\u00d7) 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]Index124DiagramCoxeter[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*323232\u00d7Semidirect subgroupsDiagramCoxeter[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}Orbifold4*36*22*322*333*22Direct subgroupsIndex248DiagramCoxeter[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)22}Orbifold64244366232223232Radical subgroupsIndex8121624DiagramCoxeter[6,4*] = [6*,4][6,4*]+ = [6*,4]+Orbifold*3333*2222223333222222See also[edit]References[edit]External links[edit]Wikimedia ErrorOur servers are currently under maintenance or experiencing a technical problem.Please try again in a few\u00a0minutes.See the error message at the bottom of this page for more\u00a0information. 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