The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
The Schläfli symbol of the order-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
The Schläfli symbol of the order-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
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.
May 06, 2024
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Order 3 6 heptagonal honeycomb Type Regular honeycomb Schlafli symbol 7 3 6 7 3 3 Coxeter diagram Cells 7 3 Faces 7 Vertex figure 3 6 Dual 6 3 7 Coxeter group 7 3 6 7 3 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 6 heptagonal 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 Contents 1 Geometry 2 Related polytopes and honeycombs 2 1 Order 3 6 octagonal honeycomb 2 2 Order 3 6 apeirogonal honeycomb 3 See also 4 References 5 External linksGeometry editThe Schlafli symbol of the order 3 6 heptagonal honeycomb is 7 3 6 with six heptagonal tilings meeting at each edge The vertex figure of this honeycomb is an triangular tiling 3 6 It has a quasiregular construction nbsp nbsp nbsp nbsp nbsp which can be seen as alternately colored cells nbsp Poincare disk model nbsp Ideal surfaceRelated polytopes and honeycombs editIt is a part of a series of regular polytopes and honeycombs with p 3 6 Schlafli symbol and triangular tiling vertex figures Hyperbolic uniform honeycombs p 3 6 and p 3 3 vte Form Paracompact Noncompact Name 3 3 6 3 3 3 4 3 6 4 3 3 5 3 6 5 3 3 6 3 6 6 3 3 7 3 6 7 3 3 8 3 6 8 3 3 3 6 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Image nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells nbsp 3 3 nbsp nbsp nbsp nbsp nbsp nbsp 4 3 nbsp nbsp nbsp nbsp nbsp nbsp 5 3 nbsp nbsp nbsp nbsp nbsp nbsp 6 3 nbsp nbsp nbsp nbsp nbsp nbsp 7 3 nbsp nbsp nbsp nbsp nbsp nbsp 8 3 nbsp nbsp nbsp nbsp nbsp nbsp 3 nbsp nbsp nbsp nbsp nbsp Order 3 6 octagonal honeycomb edit Order 3 6 octagonal honeycomb Type Regular honeycomb Schlafli symbol 8 3 6 8 3 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 8 3 nbsp Faces Octagon 8 Vertex figure triangular tiling 3 6 Dual 6 3 8 Coxeter group 8 3 6 8 3 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 6 octagonal honeycomb a regular space filling tessellation or honeycomb Each infinite cell consists of an order 6 octagonal tiling whose vertices lie on a 2 hypercycle each of which has a limiting circle on the ideal sphere The Schlafli symbol of the order 3 6 octagonal honeycomb is 8 3 6 with six octagonal tilings meeting at each edge The vertex figure of this honeycomb is a triangular tiling 3 6 It has a quasiregular construction nbsp nbsp nbsp nbsp nbsp which can be seen as alternately colored cells nbsp Poincare disk model Order 3 6 apeirogonal honeycomb edit Order 3 6 apeirogonal honeycomb Type Regular honeycomb Schlafli symbol 3 6 3 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 3 nbsp Faces Apeirogon Vertex figure triangular tiling 3 6 Dual 6 3 Coxeter group 3 6 3 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 6 apeirogonal honeycomb a regular space filling tessellation or honeycomb Each infinite cell consists of an order 3 apeirogonal tiling whose vertices lie on a 2 hypercycle each of which has a limiting circle on the ideal sphere The Schlafli symbol of the order 3 6 apeirogonal honeycomb is 3 6 with six order 3 apeirogonal tilings meeting at each edge The vertex figure of this honeycomb is a triangular tiling 3 6 nbsp Poincare disk model nbsp Ideal surface It has a quasiregular construction nbsp nbsp nbsp nbsp nbsp which can be seen as alternately colored cells See also editConvex uniform honeycombs in hyperbolic space List of regular polytopesReferences 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 6 heptagonal honeycomb amp oldid 1199789131, wikipedia, wiki, book, books, library,
