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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"
!bgcolor=#e7dcc3 colspan=2|Order-7-3 square honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{4,7,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|7|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Order-7 square tiling|{4,7}]] [[File:H2_tiling_247-4.png|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|{4}]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-7-4 triangualar honeycomb|{3,7,4}]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[4,7,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 square honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[heptagonal 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 square honeycomb'' is {4,7,3}, with three order-4 heptagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {7,3}.
{| class=wikitable
|[[File:Hyperbolic honeycomb 4-7-3 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_473_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
=== Order-7-3 pentagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Order-7-3 pentagonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{5,7,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|7|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Order-7 pentagonal tiling|{5,7}]] [[File:H2_tiling_257-4.png|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Pentagon|{5}]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-7-5 triangular honeycomb|{3,7,5}]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[5,7,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 pentagonal honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[order-7 pentagonal 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-6-3 pentagonal honeycomb'' is {5,7,3}, with three ''order-7 pentagonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {7,3}.
{| class=wikitable
|[[File:Hyperbolic honeycomb 5-7-3 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_573_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
=== Order-7-3 hexagonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Order-7-3 hexagonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{6,7,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node|7|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Order-7 hexagonal tiling|{6,7}]] [[File:H2_tiling_267-4.png|80px]]
|-
|bgcolor=#e7dcc3|Faces||[[Hexagon|{6}]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[Order-7-6 triangular honeycomb|{3,7,6}]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[6,7,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 hexagonal honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[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}.
{| class=wikitable
|[[File:Hyperbolic honeycomb 6-7-3 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_673_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
{{-}}
=== Order-6-3 apeirogonal honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Order-7-3 apeirogonal honeycomb
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{∞,7,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|7|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Order-7 apeirogonal tiling|{∞,7}]] [[File:H2_tiling_27i-1.png|80px]]
|-
|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]]||[∞,7,3]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 apeirogonal 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 {∞,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 Poincare half-space model of H3. It shows a [[Apollonian gasket]] pattern of circles inside a largest circle.
{| class=wikitable
|[[File:Hyperbolic honeycomb i-7-3 poincare.png|240px]]<BR>[[Poincaré disk model]]
|[[File:H3_i73_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface
|}
== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
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