Order-4-6 square honeycombTypeRegular honeycombSchläfli symbol{4,6,4}Coxeter diagramsCells{4,6} Faces{4}Edge figure{4}Vertex figure{6,4}Dualself-dualCoxeter group[4,6,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.
Geometry All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It a part of a sequence of regular polychora and honeycombs {p,6,p}:
Order-6-5 hexagonal honeycomb Order-6-5 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,6,5}Coxeter diagramsCells{5,6} Faces{5}Edge figure{5}Vertex figure{6,5}Dualself-dualCoxeter group[5,6,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Order-6-6 hexagonal honeycomb Order-5-6 hexagonal honeycombTypeRegular honeycombSchläfli symbols{6,6,6}{6,(6,3,6)}Coxeter diagrams = Cells{6,6} Faces{6}Edge figure{6}Vertex figure{6,6} {(6,3,6)} Dualself-dualCoxeter group[6,5,6][6,((6,3,6))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-6 hexagonal honeycomb (or 6,6,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,6,6}. It has six order-6 hexagonal tilings, {6,6}, 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 hexagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(6,3,6)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,6,6,1+] = [6,((6,3,6))].
Order-6-infinite apeirogonal honeycomb Order-6-infinite apeirogonal honeycombTypeRegular honeycombSchläfli symbols{∞,6,∞}{∞,(6,∞,6)}Coxeter diagrams ↔ Cells{∞,6} Faces{∞}Edge figure{∞}Vertex figure {6,∞} {(6,∞,6)}Dualself-dualCoxeter group[∞,6,∞][∞,((6,∞,6))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(6,∞,6)}, Coxeter diagram, , with alternating types or colors of cells.
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes Infinite-order dodecahedral honeycomb
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space ) Table III Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II) George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013) 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.