[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/order-6-4-triangular-honeycomb-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki41\/order-6-4-triangular-honeycomb-wikipedia\/","headline":"Order-6-4 triangular honeycomb – Wikipedia","name":"Order-6-4 triangular honeycomb – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular","datePublished":"2021-09-22","dateModified":"2021-09-22","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki41\/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\/b\/bd\/CDel_node_1.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/bd\/CDel_node_1.png","height":"23","width":"9"},"url":"https:\/\/wiki.edu.vn\/en\/wiki41\/order-6-4-triangular-honeycomb-wikipedia\/","wordCount":6260,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,6,4}.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsGeometry[edit]Related polytopes and honeycombs[edit]Order-6-5 triangular honeycomb[edit]Order-6-6 triangular honeycomb[edit]Order-6-infinite triangular honeycomb[edit]See also[edit]References[edit]External links[edit]Geometry[edit]It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {3,61,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1+] = [3,61,1].Related polytopes and honeycombs[edit]It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4{3,6,p} polytopesSpaceH3FormParacompactNoncompactName{3,6,3}\u00a0{3,6,4}{3,6,5}{3,6,6}… {3,6,\u221e}ImageVertexfigure{6,3}\u00a0{6,4}{6,5}{6,6}{6,\u221e}Order-6-5 triangular honeycomb[edit]In the geometry of hyperbolic 3-space, the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,6,5}. It has five triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement.Order-6-6 triangular honeycomb[edit]Order-6-6 triangular honeycombTypeRegular honeycombSchl\u00e4fli symbols{3,6,6}{3,(6,3,6)}Coxeter diagrams = Cells{3,6} Faces{3}Edge figure{6}Vertex figure{6,6} {(6,3,6)} Dual{6,6,3}Coxeter group[3,6,6][3,((6,3,6))]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-6-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,6,6}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-6 triangular tiling vertex arrangement.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {3,(6,3,6)}, Coxeter diagram,  = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1+] = [3,((6,3,6))].Order-6-infinite triangular honeycomb[edit]Order-6-infinite triangular honeycombTypeRegular honeycombSchl\u00e4fli symbols{3,6,\u221e}{3,(6,\u221e,6)}Coxeter diagrams = Cells{3,6} Faces{3}Edge figure{\u221e}Vertex figure{6,\u221e} {(6,\u221e,6)} Dual{\u221e,6,3}Coxeter group[\u221e,6,3][3,((6,\u221e,6))]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,6,\u221e}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {3,(6,\u221e,6)}, Coxeter diagram,  = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,\u221e,1+] = [3,((6,\u221e,6))].See also[edit]References[edit]Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN\u00a00-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp.\u00a0294\u2013296)The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN\u00a099-35678, ISBN\u00a00-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table IIIJeffrey R. Weeks The Shape of Space, 2nd edition ISBN\u00a00-8247-0709-5 (Chapters 16\u201317: Geometries on Three-manifolds I,II)George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]Hao Chen, Jean-Philippe Labb\u00e9, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)External links[edit]Spherical Video: {3,6,\u221e} honeycomb with parabolic M\u00f6bius transform YouTube, Roice NelsonJohn Baez, Visual insights: {7,3,3} Honeycomb (2014\/08\/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014\/08\/14)Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. 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