[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/octagonal-tiling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/octagonal-tiling-wikipedia\/","headline":"Octagonal tiling – Wikipedia","name":"Octagonal tiling – Wikipedia","description":"From Wikipedia, the free encyclopedia Octagonal tiling Poincar\u00e9 disk model of the hyperbolic plane Type Hyperbolic regular tiling Vertex configuration","datePublished":"2020-06-25","dateModified":"2020-06-25","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\/1\/11\/H2-8-3-dual.svg\/280px-H2-8-3-dual.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/11\/H2-8-3-dual.svg\/280px-H2-8-3-dual.svg.png","height":"280","width":"280"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/octagonal-tiling-wikipedia\/","about":["Wiki"],"wordCount":18993,"articleBody":"From Wikipedia, the free encyclopediaOctagonal tilingPoincar\u00e9 disk model of the hyperbolic planeTypeHyperbolic regular tilingVertex configuration83Schl\u00e4fli symbol{8,3}t{4,8}Wythoff symbol3 | 8 22 8 | 44 4 4 |Coxeter diagramSymmetry group[8,3], (*832)[8,4], (*842)[(4,4,4)], (*444)DualOrder-8 triangular tilingPropertiesVertex-transitive, edge-transitive, face-transitiveIn geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schl\u00e4fli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.Table of ContentsUniform colorings[edit]Regular maps[edit]Related polyhedra and tilings[edit]See also[edit]References[edit]External links[edit]Uniform colorings[edit]Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.Regular maps[edit]The regular map {8,3}2,0 can be seen as a 6-coloring of the {8,3} hyperbolic tiling. Within the regular map, octagons of the same color are considered the same face shown in multiple locations. The 2,0 subscripts show the same color will repeat by moving 2 steps in a straight direction following opposite edges. This regular map also has a representation as a double covering of a cube, represented by Schl\u00e4fli symbol {8\/2,3}, with 6 octagonal faces, double wrapped {8\/2}, with 24 edges, and 16 vertices. It was described by Branko Gr\u00fcnbaum in his 2003 paper Are Your Polyhedrathe Same as My Polyhedra?[1]Related polyhedra and tilings[edit]This tiling is topologically part of sequence of regular polyhedra and tilings with Schl\u00e4fli symbol {n,3}.*n32 symmetry mutation of regular tilings: {n,3}SphericalEuclideanCompact hyperb.Paraco.Noncompact hyperbolic{2,3}{3,3}{4,3}{5,3}{6,3}{7,3}{8,3}{\u221e,3}{12i,3}{9i,3}{6i,3}{3i,3}And also is topologically part of sequence of regular tilings with Schl\u00e4fli symbol {8,n}.From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.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)3V8.6.6V35.4Uniform 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 (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]  "},{"@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\/octagonal-tiling-wikipedia\/#breadcrumbitem","name":"Octagonal tiling – Wikipedia"}}]}]