Order-6-3 square honeycombTypeRegular honeycombSchläfli symbol{4,6,3}Coxeter diagramCells{4,6} Faces{4}Vertex figure{6,3}Dual{3,6,4}Coxeter group[4,6,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal 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-6-3 square honeycomb is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:
Order-6-3 pentagonal honeycomb
Order-6-3 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,6,3}Coxeter diagramCells{5,6} Faces{5}Vertex figure{6,3}Dual{3,6,5}Coxeter group[5,6,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 pentagonal 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-6-3 pentagonal honeycomb is {5,6,3}, with three order-6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Poincaré disk modelIdeal surface
Order-6-3 hexagonal honeycomb Order-6-3 hexagonal honeycombTypeRegular honeycombSchläfli symbol{6,6,3}Coxeter diagramCells{6,6} Faces{6}Vertex figure{6,3}Dual{3,6,6}Coxeter group[6,6,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 order-6-3 hexagonal honeycomb is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
Poincaré disk modelIdeal surface
Order-6-3 apeirogonal honeycomb Order-6-3 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,6,3}Coxeter diagramCells{∞,6} FacesApeirogon {∞}Vertex figure{6,3}Dual{3,6,∞}Coxeter group[∞,6,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
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.