All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tilingvertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External linksedit
John 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.
January 01, 1970
order, heptagonal, honeycomb, type, regular, honeycomb, schläfli, symbol, coxeter, diagrams, cells, faces, edge, figure, vertex, figure, dual, self, dual, coxeter, group, properties, regular, geometry, hyperbolic, space, order, heptagonal, honeycomb, regular, . Order 3 7 heptagonal honeycomb Type Regular honeycomb Schlafli symbol 7 3 7 Coxeter diagrams Cells 7 3 Faces 7 Edge figure 7 Vertex figure 3 7 Dual self dual Coxeter group 7 3 7 Properties Regular In the geometry of hyperbolic 3 space the order 3 7 heptagonal honeycomb a regular space filling tessellation or honeycomb with Schlafli symbol 7 3 7 Contents 1 Geometry 2 Related polytopes and honeycombs 2 1 Order 3 8 octagonal honeycomb 2 2 Order 3 infinite apeirogonal honeycomb 3 See also 4 References 5 External linksGeometry editAll vertices are ultra ideal existing beyond the ideal boundary with seven heptagonal tilings existing around each edge and with an order 7 triangular tiling vertex figure nbsp Poincare disk model nbsp Ideal surfaceRelated polytopes and honeycombs editIt a part of a sequence of regular polychora and honeycombs p 3 p p 3 p regular honeycombs Space S3 Euclidean E3 H3 Form Finite Affine Compact Paracompact Noncompact Name 3 3 3 4 3 4 5 3 5 6 3 6 7 3 7 8 3 8 3 Image nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells nbsp 3 3 nbsp 4 3 nbsp 5 3 nbsp 6 3 nbsp 7 3 nbsp 8 3 nbsp 3 Vertexfigure nbsp 3 3 nbsp 3 4 nbsp 3 5 nbsp 3 6 nbsp 3 7 nbsp 3 8 nbsp 3 Order 3 8 octagonal honeycomb edit Order 3 8 octagonal honeycomb Type Regular honeycomb Schlafli symbols 8 3 8 8 3 4 3 Coxeter diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 8 3 nbsp Faces 8 Edge figure 8 Vertex figure 3 8 nbsp 3 8 3 nbsp Dual self dual Coxeter group 8 3 8 8 3 4 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 8 octagonal honeycomb is a regular space filling tessellation or honeycomb with Schlafli symbol 8 3 8 It has eight octagonal tilings 8 3 around each edge All vertices are ultra ideal existing beyond the ideal boundary with infinitely many octagonal tilings existing around each vertex in an order 8 triangular tiling vertex arrangement nbsp Poincare disk model It has a second construction as a uniform honeycomb Schlafli symbol 8 3 4 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp with alternating types or colors of cells In Coxeter notation the half symmetry is 8 3 8 1 8 3 4 3 Order 3 infinite apeirogonal honeycomb edit Order 3 infinite apeirogonal honeycomb Type Regular honeycomb Schlafli symbols 3 3 3 Coxeter diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 3 nbsp Faces Edge figure Vertex figure nbsp 3 nbsp 3 3 Dual self dual Coxeter group 3 3 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 infinite apeirogonal honeycomb is a regular space filling tessellation or honeycomb with Schlafli symbol 3 It has infinitely many order 3 apeirogonal tiling 3 around each edge All vertices are ultra ideal Existing beyond the ideal boundary with infinitely many apeirogonal tilings existing around each vertex in an infinite order triangular tiling vertex arrangement nbsp Poincare disk model nbsp Ideal surface It has a second construction as a uniform honeycomb Schlafli symbol 3 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp with alternating types or colors of apeirogonal tiling cells See also editConvex uniform honeycombs in hyperbolic space List of regular polytopes Infinite order dodecahedral honeycombReferences editCoxeter Regular Polytopes 3rd ed Dover Publications 1973 ISBN 0 486 61480 8 Tables I and II Regular polytopes and honeycombs pp 294 296 The Beauty of Geometry Twelve Essays 1999 Dover Publications LCCN 99 35678 ISBN 0 486 40919 8 Chapter 10 Regular Honeycombs in Hyperbolic Space Table III Jeffrey R Weeks The Shape of Space 2nd edition ISBN 0 8247 0709 5 Chapters 16 17 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 Labbe Lorentzian Coxeter groups and Boyd Maxwell ball packings 2013 2 Visualizing Hyperbolic Honeycombs arXiv 1511 02851 Roice Nelson Henry Segerman 2015 External links editJohn 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 Retrieved from https en wikipedia org w index php title Order 3 7 heptagonal honeycomb amp oldid 1199789146, wikipedia, wiki, book, books, library,
