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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}}
|-
|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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|bgcolor=#e7dcc3|Edge figure||[[Triangle|{3}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[heptagonal tiling|{7,3}]] [[File:H2Heptagonal tiling 237-1.pngsvg|50px]]
|-
|bgcolor=#e7dcc3|Dual||Self-dual
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|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 triangular honeycomb''' (or '''3,7,3 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,3}.
 
== 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.pngjpg|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 a part of a sequence of self-dual regular honeycombs: {''p'',7,''p''}.
 
It is also a part of a sequence of regular honeycombs with [[order-7 triangular tiling]] [[cell (geometry)|cells]]: {3,7,''p''}.
 
It isa part of a sequence of regular honeycombs with [[heptagonal tiling]] [[vertex figures]]: {''p'',7,3}.
 
{{Clear}}
 
{{-}}
=== Order-7-4 triangular honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=240
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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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|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-4 triangular honeycomb''' (or '''3,7,4 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,4}.
 
It has four [[order-7 triangular tiling]]s, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an [[order-4 hexagonal tiling]] [[vertex arrangement]].
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It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,7<sup>1,1</sup>}, 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<sup>+</sup>] = [3,7<sup>1,1</sup>].
 
{{-Clear}}
 
=== Order-7-5 triangular honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=280
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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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|}
 
{{-Clear}}
 
===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
|}
{{-Clear}}
 
===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<sup>+</sup>] = [3,((7,∞,7))].
 
{{-Clear}}
 
=== Order-7-3 square honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
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|bgcolor=#e7dcc3|Properties||Regular
|}
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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|}
 
{{-Clear}}
 
=== Order-7-3 apeirogonal honeycomb===
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|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{&infin;∞,7,3}
|-
|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]]
|-
|bgcolor=#e7dcc3|Faces||[[Apeirogon]] {&infin;∞}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-7-infinite triangular honeycomb|{3,7,&infin;∞}]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[&infin;∞,7,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 apeirogonal honeycomb''' (or '''&infin;∞,7,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[order-7 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 apeirogonal tiling honeycomb is {&infin;∞,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
Line 302 ⟶ 307:
|[[File:H3_i73_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
 
===Order-7-4 square honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-7-4 square honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{4,7,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|4|node|7|node|4|node}}<BR>{{CDD|node_1|4|node|7|node|4|node_h0}} = {{CDD|node_1|4|node|split1-77|nodes}}
|-
|bgcolor=#e7dcc3|Cells||[[order-7 square tiling|{4,7}]] [[File:H2 tiling 247-4.png|60px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Order-4 heptagonal tiling|{7,4}]]
|-
|bgcolor=#e7dcc3|Dual||self-dual
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[4,7,4]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-4 square honeycomb''' (or '''4,7,4 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,7,4}.
 
All vertices are ultra-ideal (existing beyond the ideal boundary) with four [[order-5 square tiling]]s existing around each edge and with an [[order-4 heptagonal tiling]] [[vertex figure]].
 
{| class=wikitable
|[[File:Hyperbolic honeycomb 4-7-4 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_474_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
 
{{Clear}}
 
=== Order-7-5 pentagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-7-5 pentagonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{5,7,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|5|node|7|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[order-7 pentagonal tiling|{5,7}]] [[File:H2 tiling 257-1.png|60px]]
|-
|bgcolor=#e7dcc3|Faces||[[pentagon|{5}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[pentagon|{5}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Order-5 heptagonal tiling|{7,5}]]
|-
|bgcolor=#e7dcc3|Dual||self-dual
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[5,7,5]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-5 pentagonal honeycomb''' (or '''5,7,5 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {5,7,5}.
 
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-7 pentagonal tilings existing around each edge and with an [[order-5 heptagonal tiling]] [[vertex figure]].
 
{| class=wikitable
|[[File:Hyperbolic honeycomb 5-7-5 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_575_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
{{Clear}}
 
=== Order-7-6 hexagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=280
!bgcolor=#e7dcc3 colspan=2|Order-7-6 hexagonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{6,7,6}<BR>{6,(7,3,7)}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|6|node|7|node|6|node}}<BR>{{CDD|node_1|6|node|7|node|6|node_h0}} = {{CDD|node_1|6|node|split1-77|branch}}
|-
|bgcolor=#e7dcc3|Cells||[[order-5 heptagonal tiling|{6,7}]] [[File:H2 tiling 257-4.png|60px]]
|-
|bgcolor=#e7dcc3|Faces||[[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Order-6 heptagonal tiling|{7,6}]] [[File:H2 tiling 257-4.png|40px]]<BR>{(5,3,5)} [[File:H2 tiling 357-1.png|40px]]
|-
|bgcolor=#e7dcc3|Dual||self-dual
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[6,7,6]<BR>[6,((7,3,7))]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-6 hexagonal honeycomb''' (or '''6,7,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {6,7,6}. It has six [[order-7 hexagonal tiling]]s, {6,7}, 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-6 heptagonal tiling]] [[vertex arrangement]].
 
{| class=wikitable
|[[File:Hyperbolic honeycomb 6-7-6 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_676_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
 
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<sup>+</sup>] = [6,((7,3,7))].
 
{{Clear}}
 
=== Order-7-infinite apeirogonal honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-7-infinite apeirogonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{∞,7,∞}<BR>{∞,(7,∞,7)}
|-
|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|{∞,7}]] [[File:H2 tiling 27i-1.png|60px]]
|-
|bgcolor=#e7dcc3|Faces||[[Apeirogon|{∞}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[Apeirogon|{∞}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:H2 tiling 27i-4.png|40px]] [[Infinite-order heptagonal tiling|{7,∞}]]<BR>[[File:H2 tiling 77i-4.png|40px]] {(7,∞,7)}
|-
|bgcolor=#e7dcc3|Dual||self-dual
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[∞,7,∞]<BR>[∞,((7,∞,7))]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-infinite apeirogonal honeycomb''' (or '''∞,7,∞ honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {∞,7,∞}. It has infinitely many [[order-7 apeirogonal tiling]] {∞,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an [[infinite-order heptagonal tiling]] [[vertex figure]].
 
{| class=wikitable
|[[File:Hyperbolic honeycomb i-5-i poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_i5i_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
 
It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {∞,(7,∞,7)}, Coxeter diagram, {{CDD|node_1|infin|node|split1-77|branch|labelinfin}}, with alternating types or colors of cells.
 
== See also ==
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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)[httphttps://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)]]
* [[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. [http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf]
 
[[Category:Honeycombs (geometry)3-honeycombs]]
[[Category:Regular 3-honeycombs]]
Retrieved from "https://en.wikipedia.org/wiki/Order-7-3_triangular_honeycomb"