[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/order-8-triangular-tiling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/order-8-triangular-tiling-wikipedia\/","headline":"Order-8 triangular tiling – Wikipedia","name":"Order-8 triangular tiling – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia Order-8 triangular tiling Poincar\u00e9 disk model of the hyperbolic plane Type Hyperbolic regular tiling","datePublished":"2018-05-11","dateModified":"2018-05-11","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\/d\/d0\/H2-8-3-primal.svg\/280px-H2-8-3-primal.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/d0\/H2-8-3-primal.svg\/280px-H2-8-3-primal.svg.png","height":"280","width":"280"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/order-8-triangular-tiling-wikipedia\/","about":["Wiki"],"wordCount":18384,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopediaOrder-8 triangular tilingPoincar\u00e9 disk model of the hyperbolic planeTypeHyperbolic regular tilingVertex configuration38Schl\u00e4fli symbol{3,8}(3,4,3)Wythoff symbol8 | 3 24 | 3 3Coxeter diagramSymmetry group[8,3], (*832)[(4,3,3)], (*433)[(4,4,4)], (*444)DualOctagonal tilingPropertiesVertex-transitive, edge-transitive, face-transitive       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schl\u00e4fli symbol of {3,8}, having eight regular triangles around each vertex.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsUniform colorings[edit]Symmetry[edit]Related polyhedra and tilings[edit]See also[edit]References[edit]External links[edit]Uniform colorings[edit]The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:Symmetry[edit]From [(4,4,4)] symmetry, there are 15 small index subgroups (7 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. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.Small index subgroups of [(4,4,4)] (*444)Index124DiagramCoxeter[(4,4,4)][(1+,4,4,4)] = [(4,1+,4,4)] = [(4,4,1+,4)] = [(1+,4,1+,4,4)][(4+,4+,4)]Orbifold*444*42422*222222\u00d7DiagramCoxeter[(4,4+,4)][(4,4,4+)][(4+,4,4)][(4,1+,4,1+,4)][(1+,4,4,1+,4)] = Orbifold4*222*222Direct subgroupsIndex248DiagramCoxeter[(4,4,4)]+[(4,4+,4)]+ = [(4,4,4+)]+ = [(4+,4,4)]+ = [(4,1+,4,1+,4)]+ = Orbifold4444242222222Radical subgroupsIndex816DiagramCoxeter[(4,4*,4)][(4,4,4*)][(4*,4,4)][(4,4*,4)]+[(4,4,4*)]+[(4*,4,4)]+Orbifold*2222222222222222Related polyhedra and tilings[edit]  The {3,3,8} honeycomb has {3,8} vertex figures.From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.Uniform octagonal\/triangular tilings Symmetry: [8,3], (*832)[8,3]+(832)[1+,8,3](*443)[8,3+](3*4){8,3}t{8,3}r{8,3}t{3,8}{3,8}rr{8,3}s2{3,8}tr{8,3}sr{8,3}h{8,3}h2{8,3}s{3,8} or  or Uniform dualsV83V3.16.16V3.8.3.8V6.6.8V38V3.4.8.4V4.6.16V34.8V(3.4)3V8.6.6V35.4It can also be generated from the (4 3 3) hyperbolic tilings:Uniform (4,3,3) tilings Symmetry: [(4,3,3)], (*433)[(4,3,3)]+, (433)h{8,3}t0(4,3,3)r{3,8}1\/2t0,1(4,3,3)h{8,3}t1(4,3,3)h2{8,3}t1,2(4,3,3){3,8}1\/2t2(4,3,3)h2{8,3}t0,2(4,3,3)t{3,8}1\/2t0,1,2(4,3,3)s{3,8}1\/2s(4,3,3)Uniform dualsV(3.4)3V3.8.3.8V(3.4)3V3.6.4.6V(3.3)4V3.6.4.6V6.6.8V3.3.3.3.3.4Uniform (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\/order-8-triangular-tiling-wikipedia\/#breadcrumbitem","name":"Order-8 triangular tiling – Wikipedia"}}]}]