Order-3-5 heptagonal honeycombTypeRegular honeycombSchläfli symbol{7,3,5}Coxeter diagramCells{7,3} FacesHeptagon {7}Vertex figureicosahedron {3,5}Dual{5,3,7}Coxeter group[7,3,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-5 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-3-5 heptagonal honeycomb is {7,3,5}, with five heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
Poincaré disk model(vertex centered)Ideal surface
Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with {p,3,5} Schläfli symbol, and icosahedral vertex figures.
Order-3-5 octagonal honeycomb Order-3-5 octagonal honeycombTypeRegular honeycombSchläfli symbol{8,3,5}Coxeter diagramCells{8,3} FacesOctagon {8}Vertex figureicosahedron {3,5}Dual{5,3,8}Coxeter group[8,3,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-5 octagonal 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 3-5 heptagonal honeycomb is {8,3,5}, with five octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
Poincaré disk model(vertex centered)
Order-3-5 apeirogonal honeycomb Order-3-5 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,3,5}Coxeter diagramCells{∞,3} FacesApeirogon {∞}Vertex figureicosahedron {3,5}Dual{5,3,∞}Coxeter group[∞,3,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-3-5 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-3-5 apeirogonal honeycomb is {∞,3,5}, with five order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.
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.