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|[[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}}
|-
|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}.
== Geometry==
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
|}
=== Order-7-5 hexagonal 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||[[heptagonal 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 pentagonal tiling]] [[vertex figure]].
{| class=wikitable
|[[File:Hyperbolic honeycomb 5-7-5 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_555_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
{{-}}
=== Order-5-6 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))].
{{-}}
=== 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||[[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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