Order-4-4 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,4,4}{5,41,1}Coxeter diagramCells{5,4} Faces{5}Vertex figure{4,4}Dual{4,4,5}Coxeter group[5,4,4][5,41,1]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-4 pentagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal 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-4-4 pentagonal honeycomb is {5,4,4}, with four order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with {p,4,4} Schläfli symbol, and square tiling vertex figures:
Order-4-4 hexagonal honeycomb Order-4-4 hexagonal honeycombTypeRegular honeycombSchläfli symbol{6,4,4}{6,41,1}Coxeter diagramCells{6,4} Faces{6}Vertex figure{4,4}Dual{4,4,6}Coxeter group[6,4,4][6,41,1]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-4 hexagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the octagonal tiling honeycomb is {6,4,4}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
Poincaré disk modelIdeal surface
Order-4-4 apeirogonal honeycomb Order-4-4 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,4,4}{∞,41,1}Coxeter diagramCells{∞,4} Faces{∞}Vertex figure{4,4}Dual{4,4,∞}Coxeter group[∞,4,4][∞,41,1]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4-4 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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 apeirogonal tiling honeycomb is {∞,4,4}, with three order-4 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
Poincaré disk modelIdeal 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.