View source for Heptagonal tiling honeycomb

{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Heptagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{7,3,3}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|7|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Heptagonal tiling|{7,3}]] [[File:Heptagonal tiling.svg|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Heptagon]] {7}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Tetrahedron|tetrahedron]] {3,3}
|-
|bgcolor=#e7dcc3|Dual||[[Order-7 tetrahedral honeycomb|{3,3,7}]]
|-
|bgcolor=#e7dcc3|Coxeter group||[7,3,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''heptagonal tiling honeycomb''' or '''7,3,3 honeycomb''' a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[Heptagonal tiling|heptagonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.

== Geometry==
The [[Schläfli symbol]] of the heptagonal tiling honeycomb is {7,3,3}, with three heptagonal tilings meeting at each edge.  The [[Vertex figure|vertex figure]] of this honeycomb is a tetrahedron, {3,3}.

{| class=wikitable
|- align=center
|[[File:Hyperbolic honeycomb 7-3-3 poincare vc.png|240px]]<BR>[[Poincaré disk model]]<BR>(vertex centered)
|[[File:7-3-3 Hyperbolic Honeycomb Rotating.gif|240px|]]<BR>Rotating
|[[File:H3_733_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}

== Related polytopes and honeycombs ==
It is a part of a series of regular polytopes and honeycombs with {''p'',3,3} [[Schläfli symbol]], and tetrahedral vertex figures:

It is a part of a series of regular honeycombs, {7,3,''p''}.
{| class=wikitable
!{7,3,3}
![[Order-3-4 heptagonal honeycomb|{7,3,4}]]
![[Order-3-5 heptagonal honeycomb|{7,3,5}]]
![[Order-3-6 heptagonal honeycomb|{7,3,6}]]
![[Order-3-7 heptagonal honeycomb|{7,3,7}]]
!{7,3,8}
!...{7,3,∞}
|-
|[[File:Hyperbolic honeycomb 7-3-3 poincare vc.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-4 poincare vc.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-5 poincare vc.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-6 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-7 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-8 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-3-i poincare.png|80px]]
|}

It is a part of a series of regular honeycombs, with {7,''p'',3}.
{| class=wikitable
!{7,3,3}
!{7,4,3}
!{7,5,3}...
<!--!{7,6,3}
!{7,7,3}
!{7,8,3}
!{7,∞,3}-->
|-
|[[File:Hyperbolic honeycomb 7-3-3 poincare vc.png|80px]]
|[[File:Hyperbolic honeycomb 7-4-3 poincare vc.png|80px]]
|[[File:Hyperbolic honeycomb 7-5-3 poincare vc.png|80px]]
<!--|[[File:Hyperbolic honeycomb 7-6-3 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-7-3 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-8-3 poincare.png|80px]]
|[[File:Hyperbolic honeycomb 7-∞-3 poincare.png|80px]]-->
|}

=== Octagonal tiling honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Octagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{8,3,3}<BR>t{8,4,3}<BR>2t{4,8,4}<BR>t{4<sup>[3,3]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|8|node|3|node|3|node}}<BR>{{CDD|node_1|4|node_1|8|node|3|node}}<BR>{{CDD|node|8|node_1|4|node_1|8|node}}<BR>{{CDD|branch_11|split2-44|node_1|8|node}}<BR>{{CDD|label4|branch_11|splitcross|branch_11|label4}} (all 4s)
|-
|bgcolor=#e7dcc3|Cells||[[Octagonal tiling|{8,3}]] [[File:H2-8-3-dual.svg|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Tetrahedron|tetrahedron]] {3,3}
|-
|bgcolor=#e7dcc3|Dual||{3,3,8}
|-
|bgcolor=#e7dcc3|Coxeter group||[8,3,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''octagonal tiling honeycomb''' or '''8,3,3 honeycomb''' a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[Octagonal tiling|octagonal 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 octagonal tiling honeycomb is {8,3,3}, with three octagonal tilings meeting at each edge.  The [[Vertex figure|vertex figure]] of this honeycomb is an tetrahedron, {3,3}.

{| class=wikitable
|- align=center
|[[File:Hyperbolic honeycomb 8-3-3 poincare vc.png|320px]]<BR>[[Poincaré disk model]] (vertex centered)
|[[File:Hyperbolic subgroup tree 338-direct.png|240px]]<BR>Direct subgroups of [8,3,3]
|}

{{Clear}}

=== Apeirogonal tiling honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Apeirogonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{∞,3,3}<BR>t{∞,3,3}<BR>2t{∞,∞,∞}<BR>t{∞<sup>[3,3]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|infin|node|3|node|3|node}}<BR>{{CDD|node_1|infin|node_1|infin|node|3|node}}<BR>{{CDD|node|infin|node_1|infin|node_1|infin|node}}<BR>{{CDD|labelinfin|branch_11|split2-ii|node_1|infin|node}}<BR>{{CDD|labelinfin|branch_11|splitcross|branch_11|labelinfin}} (all ∞)
|-
|bgcolor=#e7dcc3|Cells||[[Order-3 apeirogonal tiling|{∞,3}]] [[File:H2-I-3-dual.svg|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Apeirogon]] {∞}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Tetrahedron|tetrahedron]] {3,3}
|-
|bgcolor=#e7dcc3|Dual||{3,3,∞}
|-
|bgcolor=#e7dcc3|Coxeter group||[∞,3,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''apeirogonal tiling honeycomb''' or '''∞,3,3 honeycomb''' a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[Apeirogonal tiling|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 {∞,3,3}, with three apeirogonal tilings meeting at each edge.  The [[Vertex figure|vertex figure]] of this honeycomb is an tetrahedron, {3,3}.

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

{| class=wikitable
|- align=center
|[[File:Hyperbolic honeycomb i-3-3 poincare vc.png|320px]]<BR>[[Poincaré disk model]] (vertex centered)
|[[File:H3_i33_UHS_plane_at_infinity.png|320px]]<BR>Ideal surface
|}

== See also ==
* Convex uniform honeycombs in hyperbolic space
* List of regular polytopes

== References ==
{{reflist}}
*Coxeter, ''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
* [[Biography: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, [[Biography:Henry Segerman|Henry Segerman]] (2015)

== External links ==
*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) 
* [[Biography:Danny Calegari|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)]]
[[Category:Heptagonal tilings]]

{{Sourceattribution|Heptagonal tiling honeycomb}}