View source for Order-4-5 square honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-4-5 square honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{4,4,5}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Uniform tiling 44-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[Pentagon|{5}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Order-5 square tiling|{4,5}]] [[File:H2-5-4-primal.svg|50px]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-4-4 pentagonal honeycomb|{5,4,4}]]
|-
|bgcolor=#e7dcc3|Coxeter group||[4,4,5]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-4-5 square honeycomb''' is a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,4,5}. It has five [[Square tiling|square tiling]] {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an [[Order-5 square tiling|order-5 square tiling]] [[Vertex arrangement|vertex arrangement]].

== Images==
{| class=wikitable width=480
|[[File:Hyperbolic honeycomb 4-4-5 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_445_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}

== Related polytopes and honeycombs ==

It a part of a sequence of regular polychora and honeycombs with [[Octahedron|square tiling]] cells: {4,4,''p''}
{{Clear}}

=== Order-4-6 square honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=280
!bgcolor=#e7dcc3 colspan=2|Order-4-6 square honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{4,4,6}<BR>{4,(4,3,4)}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node|6|node}}<BR>{{CDD|node_1|4|node|4|node|6|node_h0}} = {{CDD|node_1|4|node|split1-44|branch}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Uniform tiling 44-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[Hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Order-6 square tiling|{4,6}]] [[File:H2 tiling 246-4.png|40px]]<BR>{(4,3,4)} [[File:Uniform tiling 443-t1.png|40px]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-4-4 hexagonal honeycomb|{6,4,4}]]
|-
|bgcolor=#e7dcc3|Coxeter group||[4,4,6]<BR>[4,((4,3,4))]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-4-6 square honeycomb''' is a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,4,6}. It has six [[Square tiling|square tiling]], {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an [[Order-6 square tiling|order-6 square tiling]] [[Vertex arrangement|vertex arrangement]].

{| class=wikitable width=480
|[[File:Hyperbolic honeycomb 4-4-6 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_446_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}

It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {4,(4,3,4)}, Coxeter diagram, {{CDD|node_1|4|node|split1-44|branch}}, with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1<sup>+</sup>] = [4,((4,3,4))].

{{Clear}}

===Order-4-infinite square honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width=240
!bgcolor=#e7dcc3 colspan=2|Order-4-infinite square honeycomb
|-
|bgcolor=#e7dcc3|Type||Regular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{4,4,∞}<BR>{4,(4,∞,4)}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node|infin|node}}<BR>{{CDD|node_1|4|node|4|node|infin|node_h0}} = {{CDD|node_1|4|node|split1-44|branch|labelinfin}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Uniform tiling 44-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|Edge figure||[[Apeirogon|{∞}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[Infinite-order square tiling|{4,∞}]] [[File:H2 tiling 24i-4.png|40px]]<BR>{(4,∞,4)} [[File:H2 tiling 44i-4.png|40px]]
|-
|bgcolor=#e7dcc3|Dual||{∞,4,4}
|-
|bgcolor=#e7dcc3|Coxeter group||[∞,4,3]<BR>[4,((4,∞,4))]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-4-infinite square honeycomb''' is a regular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,4,∞}. It has infinitely many [[Square tiling|square tiling]], {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an [[Infinite-order square tiling|infinite-order square tiling]] [[Vertex arrangement|vertex arrangement]].

{| class=wikitable width=480
|[[File:Hyperbolic honeycomb 4-4-i poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_44i_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}

It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {4,(4,∞,4)}, Coxeter diagram, {{CDD|node_1|4|node|4|node|infin|node_h0}} = {{CDD|node_1|4|node|split1-44|branch|labelinfin}}, with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1<sup>+</sup>] = [4,((4,∞,4))].

== 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]) 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:Isogonal 3-honeycombs]]
[[Category:Isochoric 3-honeycombs]]
[[Category:Order-4-n 3-honeycombs]]
[[Category:Order-n-5 3-honeycombs]]
[[Category:Regular 3-honeycombs]]

{{Sourceattribution|Order-4-5 square honeycomb}}