• Source: Order-6-3 square honeycomb

      • In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 square honeycomb or 4,3/info/6" target="_blank">6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.


      Geometry


      The Schläfli symbol of the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 square honeycomb is {4,3/info/6" target="_blank">6,3}, with three 3/info/order" target="_blank">order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {3/info/6" target="_blank">6,3}.


      Related polytopes and honeycombs


      It is a part of a series of regular polytopes and honeycombs with {p,3/info/6" target="_blank">6,3} Schläfli symbol, and dodecahedral vertex figures:


      = 3/info/order" target="_blank">Order-3/info/6" target="_blank">6-3 pentagonal honeycomb

      =

      In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 pentagonal honeycomb or 5,3/info/6" target="_blank">6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">order-3/info/6" target="_blank">6 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
      The Schläfli symbol of the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 pentagonal honeycomb is {5,3/info/6" target="_blank">6,3}, with three 3/info/order" target="_blank">order-3/info/6" target="_blank">6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {3/info/6" target="_blank">6,3}.


      = 3/info/order" target="_blank">Order-3/info/6" target="_blank">6-3 hexagonal honeycomb

      =

      In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 hexagonal honeycomb or 3/info/6" target="_blank">6,3/info/6" target="_blank">6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">order-3/info/6" target="_blank">6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
      The Schläfli symbol of the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 hexagonal honeycomb is {3/info/6" target="_blank">6,3/info/6" target="_blank">6,3}, with three 3/info/order" target="_blank">order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {3/info/6" target="_blank">6,3}.


      = 3/info/order" target="_blank">Order-3/info/6" target="_blank">6-3 apeirogonal honeycomb

      =

      In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/6" target="_blank">6-3 apeirogonal honeycomb or ∞,3/info/6" target="_blank">6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">order-3/info/6" target="_blank">6 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
      The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,3/info/6" target="_blank">6,3}, with three 3/info/order" target="_blank">order-3/info/6" target="_blank">6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {3/info/6" target="_blank">6,3}.
      The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.


      See also


      Convex uniform honeycombs in hyperbolic space
      List of regular polytopes


      References



      Coxeter, 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 Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
      Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)


      External links


      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. [3]

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