[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/tetraoctagonal-tiling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/tetraoctagonal-tiling-wikipedia\/","headline":"Tetraoctagonal tiling – Wikipedia","name":"Tetraoctagonal tiling – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia Tetraoctagonal tiling Poincar\u00e9 disk model of the hyperbolic plane Type Hyperbolic uniform tiling Vertex","datePublished":"2018-05-10","dateModified":"2018-05-10","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\/3\/3b\/H2_tiling_248-2.png\/280px-H2_tiling_248-2.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3b\/H2_tiling_248-2.png\/280px-H2_tiling_248-2.png","height":"280","width":"280"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/tetraoctagonal-tiling-wikipedia\/","about":["Wiki"],"wordCount":21142,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopediaTetraoctagonal tilingPoincar\u00e9 disk model of the hyperbolic planeTypeHyperbolic uniform tilingVertex configuration(4.8)2Schl\u00e4fli symbolr{8,4} or {84}{displaystyle {begin{Bmatrix}8\\4end{Bmatrix}}}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4rr{8,8}rr(4,4,4)t0,1,2,3(\u221e,4,\u221e,4)Wythoff symbol2 | 8 4Coxeter diagram or  or Symmetry group[8,4], (*842)[8,8], (*882)[(4,4,4)], (*444)[(\u221e,4,\u221e,4)], (*4242)DualOrder-8-4 quasiregular rhombic tilingPropertiesVertex-transitive edge-transitiveIn geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsConstructions[edit]Symmetry[edit]Related polyhedra and tiling[edit]See also[edit]References[edit]External links[edit]Constructions[edit]There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(\u221e,4,\u221e,4)], (*4242).Four uniform constructions of 4.8.4.8NameTetra-octagonal tilingRhombi-octaoctagonal tilingImageSymmetry[8,4](*842)[8,8] = [8,4,1+](*882) = [(4,4,4)] = [1+,8,4](*444) = [(\u221e,4,\u221e,4)] = [1+,8,4,1+](*4242) =  or Schl\u00e4flir{8,4}rr{8,8}=r{8,4}1\/2r(4,4,4)=r{4,8}1\/2t0,1,2,3(\u221e,4,\u221e,4)=r{8,4}1\/4Coxeter =  =  =  or Symmetry[edit]The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Related polyhedra and tiling[edit]Uniform octagonal\/square tilings [8,4], (*842)(with [8,8] (*882), [(4,4,4)] (*444) , [\u221e,4,\u221e] (*4222) index 2 subsymmetries)(And [(\u221e,4,\u221e,4)] (*4242) index 4 subsymmetry)= = = = = = = = = = = {8,4}t{8,4}r{8,4}2t{8,4}=t{4,8}2r{8,4}={4,8}rr{8,4}tr{8,4}Uniform dualsV84V4.16.16V(4.8)2V8.8.8V48V4.4.4.8V4.8.16Alternations[1+,8,4](*444)[8+,4](8*2)[8,1+,4](*4222)[8,4+](4*4)[8,4,1+](*882)[(8,4,2+)](2*42)[8,4]+(842)= = = = = = h{8,4}s{8,4}hr{8,4}s{4,8}h{4,8}hrr{8,4}sr{8,4}Alternation dualsV(4.4)4V3.(3.8)2V(4.4.4)2V(3.4)3V88V4.44V3.3.4.3.8Uniform octaoctagonal tilings Symmetry: [8,8], (*882) = =  = =  = =  = =  = =  = =  = = {8,8}t{8,8}r{8,8}2t{8,8}=t{8,8}2r{8,8}={8,8}rr{8,8}tr{8,8}Uniform dualsV88V8.16.16V8.8.8.8V8.16.16V88V4.8.4.8V4.16.16Alternations[1+,8,8](*884)[8+,8](8*4)[8,1+,8](*4242)[8,8+](8*4)[8,8,1+](*884)[(8,8,2+)](2*44)[8,8]+(882) =  =  =  = =  = = h{8,8}s{8,8}hr{8,8}s{8,8}h{8,8}hrr{8,8}sr{8,8}Alternation dualsV(4.8)8V3.4.3.8.3.8V(4.4)4V3.4.3.8.3.8V(4.8)8V46V3.3.8.3.8Uniform (4,4,4) tilings Symmetry: [(4,4,4)], (*444)[(4,4,4)]+(444)[(1+,4,4,4)](*4242)[(4+,4,4)](4*22)t0(4,4,4)h{8,4}t0,1(4,4,4)h2{8,4}t1(4,4,4){4,8}1\/2t1,2(4,4,4)h2{8,4}t2(4,4,4)h{8,4}t0,2(4,4,4)r{4,8}1\/2t0,1,2(4,4,4)t{4,8}1\/2s(4,4,4)s{4,8}1\/2h(4,4,4)h{4,8}1\/2hr(4,4,4)hr{4,8}1\/2Uniform dualsV(4.4)4V4.8.4.8V(4.4)4V4.8.4.8V(4.4)4V4.8.4.8V8.8.8V3.4.3.4.3.4V88V(4,4)3See also[edit]References[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\/tetraoctagonal-tiling-wikipedia\/#breadcrumbitem","name":"Tetraoctagonal tiling – Wikipedia"}}]}]