Order-7-3 triangular honeycomb: Difference between revisions

Revision as of 19:34, 18 April 2022

Revision as of 00:14, 28 January 2024

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It isa part of a sequence of regular honeycombs with [[heptagonal tiling]] [[vertex figures]]: {''p'',7,3}.

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=== Order-7-4 triangular honeycomb===

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It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,7^1,1}, Coxeter diagram, {{CDD|node_1|3|node|split1-77|nodes}}, with alternating types or colors of order-7 triangular tiling cells. In [[Coxeter notation]] the half symmetry is [3,7,4,1⁺] = [3,7^1,1].

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=== Order-7-5 triangular honeycomb===

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===Order-7-6 triangular honeycomb===

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|[[File:H3_376_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

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===Order-7-infinite triangular honeycomb===

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It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,(7,∞,7)}, Coxeter diagram, {{CDD|node_1|3|node|7|node|infin|node_h0}} = {{CDD|node_1|3|node|split1-77|branch|labelinfin}}, with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,∞,1⁺] = [3,((7,∞,7))].

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=== Order-7-3 square honeycomb===

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=== Order-7-3 apeirogonal honeycomb===

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=== Order-7-5 pentagonal honeycomb===

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|[[File:H3_575_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

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=== Order-7-6 hexagonal honeycomb===

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It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {6,(7,3,7)}, Coxeter diagram, {{CDD|node_1|6|node|split1-77|branch}}, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,7,6,1⁺] = [6,((7,3,7))].

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=== Order-7-infinite apeirogonal honeycomb ===

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*[[H. S. M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;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 Weeks (mathematician)|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]