Order-3-6 heptagonal honeycombTypeRegular honeycombSchläfli symbol{7,3,6}{7,3}Coxeter diagram = Cells{7,3} Faces{7} Vertex figure{3,6}Dual{6,3,7}Coxeter group[7,3,6][7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-6 heptagonal 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-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.
Order-3-6 octagonal honeycomb Order-3-6 octagonal honeycombTypeRegular honeycombSchläfli symbol{8,3,6}{8,3}Coxeter diagram = Cells{8,3} FacesOctagon {8}Vertex figuretriangular tiling {3,6}Dual{6,3,8}Coxeter group[8,3,6][8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
Poincaré disk model
Order-3-6 apeirogonal honeycomb Order-3-6 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,3,6}{∞,3}Coxeter diagram = Cells{∞,3} FacesApeirogon {∞}Vertex figuretriangular tiling {3,6}Dual{6,3,∞}Coxeter group[∞,3,6][∞,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal 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-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
Poincaré disk modelIdeal surface
It has a quasiregular construction, , which can be seen as alternately colored cells.
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.