Order-6-4 triangular honeycombTypeRegular honeycombSchläfli symbols{3,6,4}Coxeter diagrams = Cells{3,6} Faces{3}Edge figure{4}Vertex figure{6,4} r{6,6} Dual{4,6,3}Coxeter group[3,6,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,4}.
Geometry It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1+] = [3,61,1].
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}
Order-6-5 triangular honeycomb Order-6-5 triangular honeycombTypeRegular honeycombSchläfli symbol{3,6,5}Coxeter diagramCells{3,6} Faces{3}Edge figure{5}Vertex figure{6,5} Dual{5,6,3}Coxeter group[3,6,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,5}. It has five triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
Order-6-6 triangular honeycomb Order-6-6 triangular honeycombTypeRegular honeycombSchläfli symbols{3,6,6}{3,(6,3,6)}Coxeter diagrams = Cells{3,6} Faces{3}Edge figure{6}Vertex figure{6,6} {(6,3,6)} Dual{6,6,3}Coxeter group[3,6,6][3,((6,3,6))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,6}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-6 triangular tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,3,6)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1+] = [3,((6,3,6))].
Order-6-infinite triangular honeycomb Order-6-infinite triangular honeycombTypeRegular honeycombSchläfli symbols{3,6,∞}{3,(6,∞,6)}Coxeter diagrams = Cells{3,6} Faces{3}Edge figure{∞}Vertex figure{6,∞} {(6,∞,6)} Dual{∞,6,3}Coxeter group[∞,6,3][3,((6,∞,6))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,∞}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,∞,6)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1+] = [3,((6,∞,6))].
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 Spherical Video: {3,6,∞} honeycomb with parabolic Möbius transform YouTube, Roice Nelson 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.