[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/order-7-3-triangular-honeycomb-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki40\/order-7-3-triangular-honeycomb-wikipedia\/","headline":"Order-7-3 triangular honeycomb – Wikipedia","name":"Order-7-3 triangular honeycomb – Wikipedia","description":"before-content-x4 In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,3 honeycomb) is a regular space-filling tessellation (or","datePublished":"2017-12-22","dateModified":"2017-12-22","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki40\/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\/wiki40\/order-7-3-triangular-honeycomb-wikipedia\/","wordCount":5811,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,7,3}.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsGeometry[edit]Related polytopes and honeycombs[edit]Order-7-4 triangular honeycomb[edit]Order-7-5 triangular honeycomb[edit]Order-7-6 triangular honeycomb[edit]Order-7-infinite triangular honeycomb[edit]Order-7-3 square honeycomb[edit]Order-7-3 pentagonal honeycomb[edit]Order-7-3 hexagonal honeycomb[edit]Order-7-3 apeirogonal honeycomb[edit]Order-7-4 square honeycomb[edit]Order-7-5 pentagonal honeycomb[edit]Order-7-6 hexagonal honeycomb[edit]Order-7-infinite apeirogonal honeycomb[edit]See also[edit]References[edit]External links[edit]Geometry[edit]It has three order-7 triangular tiling {3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in a heptagonal tiling vertex figure.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Related polytopes and honeycombs[edit]It a part of a sequence of self-dual regular honeycombs: {p,7,p}.It is a part of a sequence of regular honeycombs with order-7 triangular tiling cells: {3,7,p}.It isa part of a sequence of regular honeycombs with heptagonal tiling vertex figures: {p,7,3}.Order-7-4 triangular honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-4 triangular honeycomb (or 3,7,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,7,4}.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4It has four order-7 triangular tilings, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 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 {3,71,1}, Coxeter diagram, , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,4,1+] = [3,71,1].Order-7-5 triangular honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,7,5}. It has five order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-5 heptagonal tiling vertex figure.Order-7-6 triangular honeycomb[edit]Order-7-6 triangular honeycombTypeRegular honeycombSchl\u00e4fli symbols{3,7,6}{3,(7,3,7)}Coxeter diagrams = Cells{3,7} Faces{3}Edge figure{6}Vertex figure{7,6} {(7,3,7)} Dual{6,7,3}Coxeter group[3,7,6]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-7-6 triangular honeycomb (or 3,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,7,6}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-6 heptagonal tiling, {7,6}, vertex figure.Order-7-infinite triangular honeycomb[edit]Order-7-infinite triangular honeycombTypeRegular honeycombSchl\u00e4fli symbols{3,7,\u221e}{3,(7,\u221e,7)}Coxeter diagrams = Cells{3,7} Faces{3}Edge figure{\u221e}Vertex figure{7,\u221e} {(7,\u221e,7)} Dual{\u221e,7,3}Coxeter group[\u221e,7,3][3,((7,\u221e,7))]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-7-infinite triangular honeycomb (or 3,7,\u221e honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {3,7,\u221e}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an infinite-order heptagonal tiling, {7,\u221e}, vertex figure.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {3,(7,\u221e,7)}, Coxeter diagram,  = , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,\u221e,1+] = [3,((7,\u221e,7))].Order-7-3 square honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-3 square honeycomb (or 4,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.The Schl\u00e4fli symbol of the order-7-3 square honeycomb is {4,7,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.Order-7-3 pentagonal honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-3 pentagonal honeycomb (or 5,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.The Schl\u00e4fli symbol of the order-6-3 pentagonal honeycomb is {5,7,3}, with three order-7 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.Order-7-3 hexagonal honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-3 hexagonal honeycomb (or 6,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.The Schl\u00e4fli symbol of the order-7-3 hexagonal honeycomb is {6,7,3}, with three order-5 hexagonal tilings meeting at each edge.  The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.Order-7-3 apeirogonal honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-3 apeirogonal honeycomb (or \u221e,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.The Schl\u00e4fli symbol of the apeirogonal tiling honeycomb is {\u221e,7,3}, with three order-7 apeirogonal tilings meeting at each edge.  The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.The “ideal surface” projection below is a plane-at-infinity, in the Poincar\u00e9 half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.Order-7-4 square honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-4 square honeycomb (or 4,7,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {4,7,4}.All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 heptagonal tiling vertex figure.Order-7-5 pentagonal honeycomb[edit]In the geometry of hyperbolic 3-space, the order-7-5 pentagonal honeycomb (or 5,7,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {5,7,5}.All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-7 pentagonal tilings existing around each edge and with an order-5 heptagonal tiling vertex figure.Order-7-6 hexagonal honeycomb[edit]Order-7-6 hexagonal honeycombTypeRegular honeycombSchl\u00e4fli symbols{6,7,6}{6,(7,3,7)}Coxeter diagrams = Cells{6,7} Faces{6}Edge figure{6}Vertex figure{7,6} {(5,3,5)} Dualself-dualCoxeter group[6,7,6][6,((7,3,7))]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-7-6 hexagonal honeycomb (or 6,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {6,7,6}. It has six order-7 hexagonal tilings, {6,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 heptagonal tiling vertex arrangement.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {6,(7,3,7)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,7,6,1+] = [6,((7,3,7))].Order-7-infinite apeirogonal honeycomb[edit]Order-7-infinite apeirogonal honeycombTypeRegular honeycombSchl\u00e4fli symbols{\u221e,7,\u221e}{\u221e,(7,\u221e,7)}Coxeter diagrams \u2194 Cells{\u221e,7} Faces{\u221e}Edge figure{\u221e}Vertex figure {7,\u221e} {(7,\u221e,7)}Dualself-dualCoxeter group[\u221e,7,\u221e][\u221e,((7,\u221e,7))]PropertiesRegularIn the geometry of hyperbolic 3-space, the order-7-infinite apeirogonal honeycomb (or \u221e,7,\u221e honeycomb) is a regular space-filling tessellation (or honeycomb) with Schl\u00e4fli symbol {\u221e,7,\u221e}. It has infinitely many order-7 apeirogonal tiling {\u221e,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an infinite-order heptagonal tiling vertex figure.It has a second construction as a uniform honeycomb, Schl\u00e4fli symbol {\u221e,(7,\u221e,7)}, Coxeter diagram, , with alternating types or colors of cells.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]Hyperbolic Catacombs Carousel:  {3,7,3} honeycomb 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. [3]     (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\/wiki40\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/order-7-3-triangular-honeycomb-wikipedia\/#breadcrumbitem","name":"Order-7-3 triangular honeycomb – Wikipedia"}}]}]