[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-order-4-hexagonal-tiling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-order-4-hexagonal-tiling-wikipedia\/","headline":"Truncated order-4 hexagonal tiling – Wikipedia","name":"Truncated order-4 hexagonal tiling – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the","datePublished":"2016-06-12","dateModified":"2016-06-12","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\/9\/92\/Order-6_tetrakis_square_tiling.png\/160px-Order-6_tetrakis_square_tiling.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/92\/Order-6_tetrakis_square_tiling.png\/160px-Order-6_tetrakis_square_tiling.png","height":"160","width":"160"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-order-4-hexagonal-tiling-wikipedia\/","about":["Wiki"],"wordCount":18122,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schl\u00e4fli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsConstructions[edit]Dual tiling[edit]Related polyhedra and tiling[edit]Symmetry[edit]References[edit]See also[edit]External links[edit]Constructions[edit]There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).Dual tiling[edit]The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.Related polyhedra and tiling[edit]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}Uniform hexahexagonal tilings Symmetry: [6,6], (*662) = =  = =  = =  = =  = =  = =   == {6,6}= h{4,6}t{6,6}= h2{4,6}r{6,6}{6,4}t{6,6}= h2{4,6}{6,6}= h{4,6}rr{6,6}r{6,4}tr{6,6}t{6,4}Uniform dualsV66V6.12.12V6.6.6.6V6.12.12V66V4.6.4.6V4.12.12Alternations[1+,6,6](*663)[6+,6](6*3)[6,1+,6](*3232)[6,6+](6*3)[6,6,1+](*663)[(6,6,2+)](2*33)[6,6]+(662) =  =  = h{6,6}s{6,6}hr{6,6}s{6,6}h{6,6}hrr{6,6}sr{6,6}Symmetry[edit]  Truncated order-4 hexagonal tiling with *662 mirror linesThe dual of the tiling represents the fundamental domains of (*662) orbifold symmetry.  From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) 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 fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.Small index subgroups of [6,6] (*662)Index124DiagramCoxeter[6,6][1+,6,6] = [6,6,1+] = [6,1+,6] = [1+,6,6,1+] = [6+,6+]Orbifold*662*663*3232*333333\u00d7Direct subgroupsDiagramCoxeter[6,6+][6+,6][(6,6,2+)][6,1+,6,1+] =  = =  = [1+,6,1+,6] =  = =  = Orbifold6*32*333*33Direct subgroupsIndex248DiagramCoxeter[6,6]+[6,6+]+ = [6+,6]+ = [6,1+,6]+ = [6+,6+]+ = [1+,6,1+,6]+ =  =  = Orbifold66266332323333Radical subgroupsIndex1224DiagramCoxeter[6,6*][6*,6][6,6*]+[6*,6]+Orbifold*333333333333References[edit]See also[edit]External links[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/truncated-order-4-hexagonal-tiling-wikipedia\/#breadcrumbitem","name":"Truncated order-4 hexagonal tiling – Wikipedia"}}]}]