Order-7 tetrahedral honeycombTypeHyperbolic regular honeycombSchläfli symbols{3,3,7}Coxeter diagramsCells{3,3} Faces{3}Edge figure{7}Vertex figure{3,7} Dual{7,3,3}Coxeter group[7,3,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
Images Poincaré disk model (cell-centered)Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
Related polytopes and honeycombs
It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.
It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}. {3,3,7}{4,3,7}{5,3,7}{6,3,7}{7,3,7}{8,3,7}{∞,3,7} It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.
Order-8 tetrahedral honeycomb Order-8 tetrahedral honeycombTypeHyperbolic regular honeycombSchläfli symbols{3,3,8}{3,(3,4,3)}Coxeter diagrams = Cells{3,3} Faces{3}Edge figure{8}Vertex figure{3,8} {(3,4,3)} Dual{8,3,3}Coxeter group[3,3,8][3,((3,4,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
Poincaré disk model (cell-centered)Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1+] = [3,((3,4,3))].
Infinite-order tetrahedral honeycomb Infinite-order tetrahedral honeycombTypeHyperbolic regular honeycombSchläfli symbols{3,3,∞}{3,(3,∞,3)}Coxeter diagrams = Cells{3,3} Faces{3}Edge figure{∞}Vertex figure{3,∞} {(3,∞,3)} Dual{∞,3,3}Coxeter group[∞,3,3][3,((3,∞,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,∞}. It has infinitely many tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk model (cell-centered)Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,∞,3)}, Coxeter diagram, = , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].
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.