Order-7-3 triangular honeycomb: Difference between revisions

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|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|7|node|3|node}}

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|bgcolor=#e7dcc3|Cells||[[order-7 triangular tiling|{3,7}]] [[File:H2Order-7 tilingtriangular 237-4tiling.pngsvg|40px]]

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|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#e7dcc3|Edge figure||[[Triangle|{3}]]

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|bgcolor=#e7dcc3|Vertex figure||[[heptagonal tiling|{7,3}]] [[File:H2Heptagonal tiling 237-1.pngsvg|50px]]

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|bgcolor=#e7dcc3|Dual||Self-dual

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== Geometry==

It has three [[order-7 triangular tiling]] {3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in ana [[heptagonal tiling]] [[vertex figure]].

{| class=wikitable width=680

|[[File:Hyperbolic honeycomb 3-7-3 poincare.png|200px]]<BR>[[Poincaré disk model]]

|[[File:H3_373_UHS_plane_at_infinity.png|200px]]<BR>Ideal surface

|[[File:Order-7-3 triangular honeycomb UHS.jpg|280px]]<BR>[[Poincaré half-plane model|Upper half space model]] with selective cells shown<ref>[http://gallery.bridgesmathart.org/exhibitions/2015-joint-mathematics-meetings/roice3 Hyperbolic Catacombs] Roice Nelson and Henry Segerman, 2014</ref>

|}

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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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|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|7|node|4|node}}<BR>{{CDD|node_1|3|node|7|node|4|node_h0}} = {{CDD|node_1|3|node|split1-77|nodes}}

|-

|bgcolor=#e7dcc3|Cells||[[order-7 triangular tiling|{3,7}]] [[File:H2Order-7 tilingtriangular 237-4tiling.pngsvg|40px]]

|-

|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]

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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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|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|7|node|5|node}}

|-

|bgcolor=#e7dcc3|Cells||[[Order-7 triangular tiling|{3,7}]] [[File:H2Order-7 tilingtriangular 237-4tiling.pngsvg|40px]]

|-

|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]

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|}

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

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|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|7|node|6|node}}<BR>{{CDD|node_1|3|node|7|node|6|node_h0}} = {{CDD|node_1|3|node|split1-77|branch}}

|-

|bgcolor=#e7dcc3|Cells||[[order-7 triangular tiling|{3,7}]] [[File:H2Order-7 tilingtriangular 237-4tiling.pngsvg|40px]]

|-

|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]

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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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|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|7|node|infin|node}}<BR>{{CDD|node_1|3|node|7|node|infin|node_h0}} = {{CDD|node_1|3|node|split1-77|branch|labelinfin}}

|-

|bgcolor=#e7dcc3|Cells||[[order-7 triangular tiling|{3,7}]] [[File:H2Order-7 tilingtriangular 237-4tiling.pngsvg|40px]]

|-

|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]

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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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|bgcolor=#e7dcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 hexagonal honeycomb''' (or '''6,7,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of aan [[order-6 hexagonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.

The [[Schläfli symbol]] of the ''order-7-3 hexagonal honeycomb'' is {6,7,3}, with three order-5 hexagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {7,3}.

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

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|7|node|3|node}}

|-

|bgcolor=#e7dcc3|Cells||[[Order-7 apeirogonal tiling|{&infin;∞,7}]] [[File:H2_tiling_27i-1.png|80px]]

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|bgcolor=#e7dcc3|Faces||[[Apeirogon]] {∞}

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|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]

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|bgcolor=#e7dcc3|Dual||[[Order-7-infinite triangular honeycomb|{3,7,&infin;∞}]]

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|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[∞,7,3]

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The [[Schläfli symbol]] of the apeirogonal tiling honeycomb is {∞,7,3}, with three ''order-7 apeirogonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {7,3}.

The "ideal surface" projection below is a plane-at-infinity, in the PoincarePoincaré half-space model of H3. It shows aan [[Apollonian gasket]] pattern of circles inside a largest circle.

{| class=wikitable

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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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|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|infin|node|7|node|infin|node}}<BR>{{CDD|node_1|infin|node|7|node|infin|node_h0}} ↔ {{CDD|node_1|infin|node|split1-77|branch|labelinfin}}

|-

|bgcolor=#e7dcc3|Cells||[[Order-7 apeirogonal tiling|{&infin;∞,7}]] [[File:H2 tiling 27i-1.png|60px]]

|-

|bgcolor=#e7dcc3|Faces||[[Apeirogon|{∞}]]

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{{reflist}}

*[[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] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }}) 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]

* [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs arXiv:1511.02851] Roice Nelson, [[Henry Segerman]] (2015)

==External links==

* [https://www.youtube.com/watch?v=GRo_FQm2KRc Hyperbolic Catacombs Carousel: {3,7,3} honeycomb] [[YouTube]], Roice Nelson

*[[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:Honeycombs (geometry)3-honeycombs]]

[[Category:Regular 3-honeycombs]]