Order-3-4 heptagonal honeycombTypeRegular honeycombSchläfli symbol{7,3,4}Coxeter diagram = Cells{7,3} Facesheptagon {7}Vertex figureoctahedron {3,4}Dual{4,3,7}Coxeter group[7,3,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Geometry The Schläfli symbol of the order-3-4 heptagonal honeycomb is {7,3,4}, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
Poincaré disk model(vertex centered)One hyperideal cell limits to a circle on the ideal surfaceIdeal surface
Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:
Order-3-4 octagonal honeycomb Order-3-4 octagonal honeycombTypeRegular honeycombSchläfli symbol{8,3,4}Coxeter diagram = Cells{8,3} Facesoctagon {8}Vertex figureoctahedron {3,4}Dual{4,3,8}Coxeter group[8,3,4][8,31,1]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-3-4 octagonal honeycomb is {8,3,4}, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
Poincaré disk model(vertex centered)
Order-3-4 apeirogonal honeycomb Order-3-4 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,3,4}Coxeter diagram = Cells{∞,3} Facesapeirogon {∞}Vertex figureoctahedron {3,4}Dual{4,3,∞}Coxeter group[∞,3,4][∞,31,1]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-4 apeirogonal honeycomb or ∞,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
Poincaré disk model(vertex centered)Ideal surface
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.