Order-infinite-3 triangular honeycomb: Difference between revisions

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|bgcolor=#efdcc3|Edge figure||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Vertex figure||[[Order-3 apeirogonal tiling|{∞,3}]] [[File:H2 tiling 23i-1I-3-dual.pngsvg|50px]]

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|bgcolor=#efdcc3|Dual||Self-dual

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 triangular honeycomb''' (or '''3,∞,3 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,3}.

== Geometry==

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It is a part of a sequence of self-dual regular honeycombs: {''p'',∞,''p''}.

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=== Order-infinite-4 triangular honeycomb===

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-4 triangular honeycomb''' (or '''3,∞,4 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,4}.

It has four [[infinite-order triangular tiling]]s, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an [[order-4 apeirogonal tiling]] [[vertex figure]].

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

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=== Order-infinite-5 triangular honeycomb===

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===Order-infinite-6 triangular honeycomb===

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|[[File:H3_3i6_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

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

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|[[File:H3_3i7_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

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

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

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=== Order-infinite-3 square honeycomb===

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 hexagonal honeycomb''' (or '''6,∞,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of aan [[order-3 apeirogonal 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-infinite-3 hexagonal honeycomb'' is {6,∞,3}, with three infinite-order hexagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is aan order-3 apeirogonal tiling, {∞,3}.

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=== Order-infinite-3 heptagonal honeycomb===

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 heptagonal honeycomb''' (or '''7,∞,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of aan [[infinite-order heptagonal 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-infinite-3 heptagonal honeycomb'' is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is aan order-3 apeirogonal tiling, {∞,3}.

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=== Order-infinite-3 apeirogonal honeycomb===

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

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

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=== Order-infinite-5 pentagonal honeycomb===

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|[[File:Hyperbolic honeycomb 5-i-5 poincare.png|240px]]<BR>[[Poincaré disk model]]

|[[File:H3_555_UHS_plane_at_infinityH3_5i5_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

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=== Order-infinite-6 hexagonal honeycomb===

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

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=== Order-infinite-7 heptagonal honeycomb===

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

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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]

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* [https://www.youtube.com/watch?v=GRo_FQm2KRc Hyperbolic Catacombs Carousel: {3,∞,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:HoneycombsRegular (geometry)3-honeycombs]]