Order-3-4 heptagonal honeycomb

Order-3-4 heptagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{7,3,4}
Coxeter diagram	File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
Cells	{7,3} File:Heptagonal tiling.svg
Faces	heptagon {7}
Vertex figure	octahedron {3,4}
Dual	{4,3,7}
Coxeter group	[7,3,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 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-3-4 heptagonal honeycomb is {7,3,4}, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

File:Hyperbolic honeycomb 7-3-4 poincare vc.png Poincaré disk model (vertex centered)

File:Order-3-4 heptagonal honeycomb cell.png One hyperideal cell limits to a circle on the ideal surface

File:H3 734 UHS plane at infinity.png Ideal surface

Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:

Order-3-4 octagonal honeycomb

Order-3-4 octagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{8,3,4}
Coxeter diagram	File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png File:CDel node.pngFile:CDel ultra.pngFile:CDel node 1.pngFile:CDel split1-44.pngFile:CDel branch 11.pngFile:CDel label4.pngFile:CDel uaub.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel ultra.pngFile:CDel node 1.pngFile:CDel split1-44.pngFile:CDel branch 11.pngFile:CDel label4.pngFile:CDel uaub.pngFile:CDel nodes 11.png
Cells	{8,3} File:H2-8-3-dual.svg
Faces	octagon {8}
Vertex figure	octahedron {3,4}
Dual	{4,3,8}
Coxeter group	[8,3,4] [8,31,1]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 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-3-4 octagonal honeycomb is {8,3,4}, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

File:Hyperbolic honeycomb 8-3-4 poincare vc.png Poincaré disk model (vertex centered)

Order-3-4 apeirogonal honeycomb

Order-3-4 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{∞,3,4}
Coxeter diagram	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png File:CDel node.pngFile:CDel ultra.pngFile:CDel node 1.pngFile:CDel split1-ii.pngFile:CDel branch 11.pngFile:CDel labelinfin.pngFile:CDel uaub.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel ultra.pngFile:CDel node 1.pngFile:CDel split1-ii.pngFile:CDel branch 11.pngFile:CDel labelinfin.pngFile:CDel uaub.pngFile:CDel nodes 11.png
Cells	{∞,3} File:H2-I-3-dual.svg
Faces	apeirogon {∞}
Vertex figure	octahedron {3,4}
Dual	{4,3,∞}
Coxeter group	[∞,3,4] [∞,31,1]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-4 apeirogonal honeycomb or ∞,3,4 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-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

File:Hyperbolic honeycomb i-3-4 poincare vc.png Poincaré disk model (vertex centered)

File:H3 i34 UHS plane at infinity.png 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, LCCN 99-35678, (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) 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) [1]
• Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
• 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. [3]