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In the geometry of hyperbolic 3-space, the order-3-6 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.

Order-3-6 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,6}
{7,3[3]}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {7,3} Heptagonal tiling.svg
Faces {7}
Vertex figure {3,6}
Dual {6,3,7}
Coxeter group [7,3,6]
[7,3[3]]
Properties Regular

Geometry

The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an triangular tiling, {3,6}.

It has a quasiregular construction, CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel branch.png, which can be seen as alternately colored cells.

Hyperbolic honeycomb 7-3-6 poincare.png
Poincaré disk model
H3 736 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,3,6} Schläfli symbol, and triangular tiling vertex figures.

Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}
Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}
{4,3,6}
{4,3[3]}
{5,3,6}
{5,3[3]}
{6,3,6}
{6,3[3]}
{7,3,6}
{7,3[3]}
{8,3,6}
{8,3[3]}
... {∞,3,6}
{∞,3[3]}
CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel p.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.png
Image H3 336 CC center.png H3 436 CC center.png H3 536 CC center.png H3 636 FC boundary.png Hyperbolic honeycomb 7-3-6 poincare.png Hyperbolic honeycomb 8-3-6 poincare.png Hyperbolic honeycomb i-3-6 poincare.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.svg
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Heptagonal tiling.svg
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2-I-3-dual.svg
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

Order-3-6 octagonal honeycomb

Order-3-6 octagonal honeycomb
Type Regular honeycomb
Schläfli symbol {8,3,6}
{8,3[3]}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {8,3} H2-8-3-dual.svg
Faces Octagon {8}
Vertex figure triangular tiling {3,6}
Dual {6,3,8}
Coxeter group [8,3,6]
[8,3[3]]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.

It has a quasiregular construction, CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.png, which can be seen as alternately colored cells.
Hyperbolic honeycomb 8-3-6 poincare.png
Poincaré disk model

Order-3-6 apeirogonal honeycomb

Order-3-6 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,3,6}
{∞,3[3]}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {∞,3} H2-I-3-dual.svg
Faces Apeirogon {∞}
Vertex figure triangular tiling {3,6}
Dual {6,3,∞}
Coxeter group [∞,3,6]
[∞,3[3]]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-6 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-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.

Hyperbolic honeycomb i-3-6 poincare.png
Poincaré disk model
H3 i36 UHS plane at infinity.png
Ideal surface

It has a quasiregular construction, CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.png, which can be seen as alternately colored cells.
See also

Convex uniform honeycombs in hyperbolic space
List of regular polytopes

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (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]

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