Editing Convex uniform honeycomb (section)

=== Ã<sub>3</sub>, [3<sup>[4]</sup>] group ===
There are 5 forms<ref>[http://mathworld.wolfram.com/Necklace.html], [http://oeis.org/A000029 A000029] 6-1 cases, skipping one with zero marks</ref> constructed from the <math>{\tilde{A}}_3</math>, [3<sup>[4]</sup>] [[Coxeter group]], of which only the ''quarter cubic honeycomb'' is unique. There is one index 2 subgroup [3<sup>[4]</sup>]<sup>+</sup> which generates the snub form, which is not uniform, but included for completeness.

{{A3 honeycombs}}

{|class="wikitable"
|+ {{Brackets|3<sup>{{Bracket|4}}</sup>}} uniform honeycombs, [[space group]] Fd{{overline|3}}m (227)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<br/>{{CDD|branch_c1-2|3ab|branch_c1-2}}
!colspan=2|Cells by location<br/>(and count around each vertex)
!rowspan=2|Solids<br/>(Partial)
!rowspan=2|Frames<br/>(Perspective)
!rowspan=2|[[vertex figure]]
|-
!(0,1)<br/>{{CDD|nodeb|3b|branch}}
!(2,3)<br/>{{CDD|branch|3a|nodea}}
|- 
|J<sub>25,33</sub><br/>A<sub>13</sub><br/>W<sub>10</sub><br/>G<sub>6</sub><br/>q&delta;<sub>4</sub><br/>O<sub>27</sub>
|[[quarter cubic honeycomb|quarter cubic]] (cytatoh)<br/>{{CDD|branch_10r|3ab|branch_10l}} ↔ {{CDD|node_h1|4|node|3|node|4|node_h1}}<br/>q{4,3,4}
|[[File:Tetrahedron.png|30px]] (2)<br/>[[tetrahedron|(3.3.3)]]
|[[File:Truncated tetrahedron.png|30px]] (6)<br/>[[truncated tetrahedron|(3.6.6)]] 
|[[File:Quarter cubic honeycomb.png|75px]]
|[[File:Bitruncated alternated cubic tiling.png|75px]]
|[[File:t01 quarter cubic honeycomb verf.png|75px]]<br/>triangular antiprism
|}

{|class="wikitable"
|+ <[3<sup>[4]</sup>]> ↔ [4,3<sup>1,1</sup>] uniform honeycombs, [[space group]] Fm{{overline|3}}m (225)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<br/>{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c3}} ↔ {{CDD|node|3|node_c3|split1|nodeab_c1-2}}
!colspan=3|Cells by location<br/>(and count around each vertex)
!rowspan=2|Solids<br/>(Partial)
!rowspan=2|Frames<br/>(Perspective)
!rowspan=2|[[vertex figure]]
|-
!0
!(1,3)
!2
|- BGCOLOR="#e0f0e0"
|J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>h&delta;<sub>4</sub><br/>O<sub>21</sub>
|[[Tetrahedral-octahedral honeycomb|alternated cubic]] (octet)<br/>{{CDD|node_1|split1|nodes|split2|node}} ↔ {{CDD|nodes_10ru|split2|node|4|node}} ↔ {{CDD|node_h1|4|node|3|node|4|node}}<br/>h{4,3,4}
|
|[[File:Uniform polyhedron-33-t0.png|30px]] (8)<br/>[[Tetrahedron|(3.3.3)]]
|[[File:Uniform polyhedron-33-t1.svg|30px]] (6)<br/>[[Octahedron|(3.3.3.3)]]
|[[File:Tetrahedral-octahedral honeycomb2.png|75px]]
|[[File:Alternated cubic tiling.png|75px]]
|[[File:Alternated cubic honeycomb verf.svg|75px]]<br/>[[cuboctahedron]]
|- BGCOLOR="#e0f0e0"
|J<sub>22,34</sub><br/>A<sub>21</sub><br/>W<sub>17</sub><br/>G<sub>10</sub><br/>h<sub>2</sub>&delta;<sub>4</sub><br/>O<sub>25</sub>
|[[Cantic cubic honeycomb|cantic cubic]] (tatoh)<br/>{{CDD|node_1|split1|nodes_11|split2|node}} ↔ {{CDD|nodes_10ru|split2|node_1|4|node}} ↔ {{CDD|node_h1|4|node|3|node_1|4|node}}<br/>h<sub>2</sub>{4,3,4}
|[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]]
|[[File:Uniform polyhedron-33-t02.svg|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-33-t012.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]]
|[[File:Truncated Alternated Cubic Honeycomb2.png|75px]]
|[[File:Truncated alternated cubic tiling.png|75px]]
|[[File:t012 quarter cubic honeycomb verf.png|75px]]<br/>Rectangular pyramid
|}

{|class="wikitable"
|+ [2[3<sup>[4]</sup>]] ↔ [4,3,4] uniform honeycombs, [[space group]] Pm{{overline|3}}m (221)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<br/>{{CDD|node_c1|split1|nodeab_c2|split2|node_c1}} ↔ {{CDD|node|4|node_c1|3|node_c2|4|node}}
!colspan=2|Cells by location<br/>(and count around each vertex)
!rowspan=2|Solids<br/>(Partial)
!rowspan=2|Frames<br/>(Perspective)
!rowspan=2|[[vertex figure]]
|-
!(0,2)<br/>{{CDD|nodeb|3b|branch}}
!(1,3)<br/>{{CDD|branch|3b|nodeb}}
|- BGCOLOR="#a0f0a0"
|J<sub>12,32</sub><br/>A<sub>15</sub><br/>W<sub>14</sub><br/>G<sub>7</sub><br/>t<sub>1</sub>&delta;<sub>4</sub><br/>O<sub>1</sub>
|[[rectified cubic honeycomb|rectified cubic]] (rich)<br/>{{CDD|node_1|split1|nodes|split2|node_1}} ↔ {{CDD|nodes|split2|node_1|4|node}} ↔ {{CDD|nodes_11|split2|node|4|node}} ↔ {{CDD|node|4|node|3|node_1|4|node}}<br/>r{4,3,4}
|[[File:Uniform polyhedron-33-t02.svg|30px]] (2)<br/>[[cuboctahedron|(3.4.3.4)]]
|[[File:Uniform polyhedron-33-t1.svg|30px]] (1)<br/>[[octahedron|(3.3.3.3)]]
|[[File:Rectified cubic honeycomb2.png|75px]]
|[[File:Rectified cubic tiling.png|75px]]
|[[File:t02 quarter cubic honeycomb verf.png|75px]]<br/>[[cuboid]]

|}

{|class="wikitable"
|+ [4[3<sup>[4]</sup>]] ↔ {{brackets|4,3,4}} uniform honeycombs, [[space group]] Im{{overline|3}}m (229)
|-
!rowspan=2|Referenced<br/>indices
!rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<br/>{{CDD|node_c1|split1|nodeab_c1|split2|node_c1}} ↔ {{CDD|nodeab_c1|split2|node_c1|4|node_h0}} ↔ {{CDD|node_h0|4|node_c1|3|node_c1|4|node_h0}}
!colspan=2|Cells by location<br/>(and count around each vertex)
!rowspan=2|Solids<br/>(Partial)
!rowspan=2|Frames<br/>(Perspective)
!rowspan=2|[[vertex figure]]
|-
!(0,1,2,3)<br/>{{CDD|node|3|node|3|node}}
!Alt
|- BGCOLOR="#60f060"
|J<sub>16</sub><br/>A<sub>3</sub><br/>W<sub>2</sub><br/>G<sub>28</sub><br/>t<sub>1,2</sub>&delta;<sub>4</sub><br/>O<sub>16</sub>
|[[bitruncated cubic honeycomb|bitruncated cubic]] (batch)<br/>{{CDD|node_1|split1|nodes_11|split2|node_1}} ↔ {{CDD|nodes_11|split2|node_1|4|node_h0}} ↔ {{CDD|node_h0|4|node_1|3|node_1|4|node_h0}}<br/>2t{4,3,4}
|[[File:Uniform polyhedron-33-t012.png|30px]] (4)<br/>[[truncated octahedron|(4.6.6)]]
|
|[[File:Bitruncated cubic honeycomb2.png|75px]]
|[[File:Bitruncated cubic tiling.png|75px]]
|[[File:t0123 quarter cubic honeycomb verf.png|75px]]<br/>isosceles [[tetrahedron]]
|- BGCOLOR="#d0f0f0"
|Nonuniform<sub>a</sub>
|[[Bitruncated cubic honeycomb#Related polyhedra and honeycombs|Alternated cantitruncated cubic]] (bisch)<br/>{{CDD|node_h|split1|nodes_hh|split2|node_h}} ↔ {{CDD|nodes_hh|split2|node_h|4|node_h0}} ↔ {{CDD|node_h0|4|node_h|3|node_h|4|node_h0}}<br/>h2t{4,3,4}
|[[File:Uniform polyhedron-33-s012.png|30px]] (4)<br/>[[Regular icosahedron|(3.3.3.3.3)]]
|[[File:Uniform polyhedron-33-t0.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]]
|&nbsp;
|
|[[File:Alternated bitruncated cubic honeycomb verf.png|75px]]

|}