Order-4-5 square honeycombTypeRegular honeycombSchläfli symbols{4,4,5}Coxeter diagramsCells{4,4} Faces{4}Edge figure{5}Vertex figure{4,5} Dual{5,4,4}Coxeter group[4,4,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.
Images Poincaré disk modelIdeal surface
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}
Order-4-6 square honeycomb Order-4-6 square honeycombTypeRegular honeycombSchläfli symbols{4,4,6}{4,(4,3,4)}Coxeter diagrams = Cells{4,4} Faces{4}Edge figure{6}Vertex figure{4,6} {(4,3,4)} Dual{6,4,4}Coxeter group[4,4,6][4,((4,3,4))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-6 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,6}. It has six square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-6 square tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,3,4)}, Coxeter diagram, , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1+] = [4,((4,3,4))].
Order-4-infinite square honeycomb Order-4-infinite square honeycombTypeRegular honeycombSchläfli symbols{4,4,∞}{4,(4,∞,4)}Coxeter diagrams = Cells{4,4} Faces{4}Edge figure{∞}Vertex figure{4,∞} {(4,∞,4)} Dual{∞,4,4}Coxeter group[∞,4,3][4,((4,∞,4))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-infinite square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,∞}. It has infinitely many square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling 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 {4,(4,∞,4)}, Coxeter diagram, = , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1+] = [4,((4,∞,4))].
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes
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.