Order-5-3 square honeycombTypeRegular honeycombSchläfli symbol{4,5,3}Coxeter diagramCells{4,5} Faces{4}Vertex figure{5,3}Dual{3,5,4}Coxeter group[4,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 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-5-3 square honeycomb is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
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,5,3} Schläfli symbol, and dodecahedral vertex figures:
Order-5-3 pentagonal honeycomb
Order-5-3 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,5,3}Coxeter diagramCells{5,5} Faces{5}Vertex figure{5,3}Dual{3,5,5}Coxeter group[5,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 pentagonal honeycomb or 5,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 pentagonal honeycomb is {5,5,3}, with three order-5 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Poincaré disk model(Vertex centered)Ideal surface
Order-5-3 hexagonal honeycomb Order-5-3 hexagonal honeycombTypeRegular honeycombSchläfli symbol{6,5,3}Coxeter diagramCells{6,5} Faces{6}Vertex figure{5,3}Dual{3,5,6}Coxeter group[6,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 hexagonal honeycomb or 6,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 hexagonal honeycomb is {6,5,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Poincaré disk model(Vertex centered)Ideal surface
Order-5-3 heptagonal honeycomb Order-5-3 heptagonal honeycombTypeRegular honeycombSchläfli symbol{7,5,3}Coxeter diagramCells{7,5} Faces{7}Vertex figure{5,3}Dual{3,5,7}Coxeter group[7,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 heptagonal 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-5-3 heptagonal honeycomb is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Poincaré disk model(Vertex centered)Ideal surface
Order-5-3 octagonal honeycomb Order-5-3 octagonal honeycombTypeRegular honeycombSchläfli symbol{8,5,3}Coxeter diagramCells{8,5} Faces{8}Vertex figure{5,3}Dual{3,5,8}Coxeter group[8,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 octagonal honeycomb is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
Poincaré disk model(Vertex centered)
Order-5-3 apeirogonal honeycomb Order-5-3 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,5,3}Coxeter diagramCells{∞,5} FacesApeirogon {∞}Vertex figure{5,3}Dual{3,5,∞}Coxeter group[∞,5,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-3 apeirogonal honeycomb or ∞,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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 {∞,5,3}, with three order-5 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
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 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.