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Order-3-7 hexagonal honeycomb

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!bgcolor=#e7dcc3 colspan=2|Order-3-7 hexagonal honeycomb

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Poincaré disk model
bgcolor=#e7dcc3|TypeRegular honeycomb
bgcolor=#e7dcc3|Schläfli symbol{6,3,7}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node_1|6|node|3|node|7|node}}
bgcolor=#e7dcc3|Cells{6,3} 40px
bgcolor=#e7dcc3|Faces{6}
bgcolor=#e7dcc3|Edge figure{7}
bgcolor=#e7dcc3|Vertex figure{3,7}
bgcolor=#e7dcc3|Dual{7,3,6}
bgcolor=#e7dcc3|Coxeter group[6,3,7]
bgcolor=#e7dcc3|PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

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|+ Ideal surface

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Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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Closeup

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.

{{hexagonal tiling cell tessellations}}

= Order-3-8 hexagonal honeycomb=

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!bgcolor=#e7dcc3 colspan=2|Order-3-8 hexagonal honeycomb

bgcolor=#e7dcc3|TypeRegular honeycomb
bgcolor=#e7dcc3|Schläfli symbols{6,3,8}
{6,(3,4,3)}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node_1|6|node|3|node|8|node}}
{{CDD|node_1|6|node|3|node|8|node_h0}} = {{CDD|node_1|6|node|split1|branch|label4}}
bgcolor=#e7dcc3|Cells{6,3} 40px
bgcolor=#e7dcc3|Faces{6}
bgcolor=#e7dcc3|Edge figure{8}
bgcolor=#e7dcc3|Vertex figure{3,8} {(3,4,3)}
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bgcolor=#e7dcc3|Dual{8,3,6}
bgcolor=#e7dcc3|Coxeter group[6,3,8]
[6,((3,4,3))]
bgcolor=#e7dcc3|PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.

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Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, {{CDD|node_1|6|node|split1|branch|label4}}, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].

{{Clear}}

= Order-3-infinite hexagonal honeycomb =

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!bgcolor=#e7dcc3 colspan=2|Order-3-infinite hexagonal honeycomb

bgcolor=#e7dcc3|TypeRegular honeycomb
bgcolor=#e7dcc3|Schläfli symbols{6,3,∞}
{6,(3,∞,3)}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node_1|6|node|3|node|infin|node}}
{{CDD|node_1|6|node|3|node|infin|node_h0}} ↔ {{CDD|node_1|6|node|split1|branch|labelinfin}}
{{CDD|node_1|6|node_g|3sg|node_g|infin|node}} ↔ File:CDD_6-3star-infin.png
bgcolor=#e7dcc3|Cells{6,3} 40px
bgcolor=#e7dcc3|Faces{6}
bgcolor=#e7dcc3|Edge figure{∞}
bgcolor=#e7dcc3|Vertex figure{3,∞}, {(3,∞,3)}
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bgcolor=#e7dcc3|Dual{∞,3,6}
bgcolor=#e7dcc3|Coxeter group[6,3,∞]
[6,((3,∞,3))]
bgcolor=#e7dcc3|PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.

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Poincaré disk model

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Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, {{CDD|node_1|6|node|split1|branch|labelinfin}}, with alternating types or colors of hexagonal tiling cells.

See also

References

{{reflist}}

  • 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|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf 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) [http://www.sciencedirect.com/science/article/pii/0021869382903180]
  • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[https://arxiv.org/abs/1310.8608]
  • [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs arXiv:1511.02851] Roice Nelson, Henry Segerman (2015)

External links

  • John Baez, Visual insights: [http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/ {7,3,3} Honeycomb] (2014/08/01) [http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ {7,3,3} Honeycomb Meets Plane at Infinity] (2014/08/14)
  • Danny Calegari, [http://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/Kleinian Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination] 4 March 2014. [https://web.archive.org/web/20161109004910/http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf]

Category:Hexagonal tilings

Category:Infinite-order tilings

Category:3-honeycombs

Category:Regular 3-honeycombs