Order-3-7 heptagonal honeycombTypeRegular honeycombSchläfli symbol{7,3,7}Coxeter diagramsCells{7,3} Faces{7}Edge figure{7}Vertex figure{3,7}Dualself-dualCoxeter group[7,3,7]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,3,7}.
Geometry All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It a part of a sequence of regular polychora and honeycombs {p,3,p}:
Order-3-8 octagonal honeycomb Order-3-8 octagonal honeycombTypeRegular honeycombSchläfli symbols{8,3,8}{8,(3,4,3)}Coxeter diagrams = Cells{8,3} Faces{8}Edge figure{8}Vertex figure{3,8} {(3,8,3)} Dualself-dualCoxeter group[8,3,8][8,((3,4,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {8,3,8}. It has eight octagonal tilings, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.
Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].
Order-3-infinite apeirogonal honeycomb Order-3-infinite apeirogonal honeycombTypeRegular honeycombSchläfli symbols{∞,3,∞}{∞,(3,∞,3)}Coxeter diagrams ↔ Cells{∞,3} Faces{∞}Edge figure{∞}Vertex figure {3,∞} {(3,∞,3)}Dualself-dualCoxeter group[∞,3,∞][∞,((3,∞,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,3,∞}. It has infinitely many order-3 apeirogonal tiling {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal 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,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling 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.