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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:
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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:
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|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
{| 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.
|}
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It isa part of a sequence of regular honeycombs with [[heptagonal tiling]] [[vertex figures]]: {''p'',7,3}.
{{
=== 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:
|-
|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>].
{{
=== 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:
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]
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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:
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|{3}]]
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|[[File:H3_376_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
{{
===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:
|-
|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))].
{{
=== 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
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|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|7|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Order-7 apeirogonal tiling|{
|-
|bgcolor=#e7dcc3|Faces||[[Apeirogon]] {
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-7-infinite triangular honeycomb|{3,7,
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 apeirogonal honeycomb''' (or '''
The [[Schläfli symbol]] of the apeirogonal tiling honeycomb is {
The "ideal surface" projection below is a plane-at-infinity, in the
{| class=wikitable
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|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]]
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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]].
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|}
{{Clear}}
=== Order-7-5 hexagonal honeycomb===▼
=== Order-7-5 pentagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-7-5 pentagonal honeycomb
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|bgcolor=#e7dcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|5|node|7|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[
|-
|bgcolor=#e7dcc3|Faces||[[pentagon|{5}]]
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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
{| class=wikitable
|[[File:Hyperbolic honeycomb 5-7-5 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:
|}
{{
▲=== Order-5-6 hexagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=280
!bgcolor=#e7dcc3 colspan=2|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<sup>+</sup>] = [6,((7,3,7))].
{{
=== 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|{
|-
|bgcolor=#e7dcc3|Faces||[[Apeirogon|{∞}]]
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|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-infinite apeirogonal honeycomb''' (or '''
{| class=wikitable
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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. 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/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]▼
▲* [[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:
[[Category:Regular 3-honeycombs]]
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