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Convex uniform honeycomb
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=== BΜ<sub>3</sub>, [4,3<sup>1,1</sup>] group === The <math>{\tilde{B}}_3</math>, [4,3] group offers 11 derived forms via truncation operations, four being unique uniform honeycombs. There are 3 index 2 subgroups that generate alternations: [1<sup>+</sup>,4,3<sup>1,1</sup>], [4,(3<sup>1,1</sup>)<sup>+</sup>], and [4,3<sup>1,1</sup>]<sup>+</sup>. The first generates repeated honeycomb, and the last two are nonuniform but included for completeness. The honeycombs from this group are called ''alternated cubic'' because the first form can be seen as a ''cubic honeycomb'' with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps. Nodes are indexed left to right as ''0,1,0',3'' with 0' being below and interchangeable with ''0''. The ''alternate cubic'' names given are based on this ordering. {{B3 honeycombs}} {|class="wikitable" |+ [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 !colspan=4|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|[[vertex figure]] |- !(0)<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(1)<br/>{{CDD|nodea|2|nodeb|2|nodea}} !(0')<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(3)<br/>{{CDD|nodea|3a|branch}} |- |J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>hδ<sub>4</sub><br/>O<sub>21</sub> |[[Tetrahedral-octahedral honeycomb|Alternated cubic]] (octet)<br/>{{CDD|nodes_10ru|split2|node|4|node}} β {{CDD|node_h1|4|node|3|node|4|node}} | | |[[File:Octahedron.png|30px]] (6)<br/>[[Octahedron|(3.3.3.3)]] |[[File:Tetrahedron.png|30px]](8)<br/>[[Tetrahedron|(3.3.3)]] |[[File:Tetrahedral-octahedral honeycomb.png|76px]] |[[File:Alternated cubic tiling.png|75px]] |[[File:Alternated cubic honeycomb verf.svg|60px]]<br/>[[cuboctahedron]] |- |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>δ<sub>4</sub><br/>O<sub>25</sub> |[[Cantic cubic honeycomb|Cantic cubic]] (tatoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node}} β {{CDD|node_h1|4|node|3|node_1|3|node}} |[[File:Cuboctahedron.png|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]] | |[[File:Truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]] |[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]] ||[[File:Truncated Alternated Cubic Honeycomb.svg|75px]] |[[File:Truncated alternated cubic tiling.png|75px]] |[[File:Truncated alternated cubic honeycomb verf.png|60px]]<br/>rectangular pyramid |- |J<sub>23</sub><br/>A<sub>16</sub><br/>W<sub>11</sub><br/>G<sub>5</sub><br/>h<sub>3</sub>δ<sub>4</sub><br/>O<sub>26</sub> |[[Runcic cubic honeycomb|Runcic cubic]] (sratoh)<br/>{{CDD|nodes_10ru|split2|node|4|node_1}} β {{CDD|node_h1|4|node|3|node|4|node_1}} |[[File:hexahedron.png|30px]] (1)<br/>[[cube]] | |[[File:small rhombicuboctahedron.png|30px]] (3)<br/>[[rhombicuboctahedron|(3.4.4.4)]] |[[File:tetrahedron.png|30px]] (1)<br/>[[tetrahedron|(3.3.3)]] |[[File:Runcinated alternated cubic honeycomb.jpg|75px]] |[[File:Runcinated alternated cubic tiling.png|75px]] |[[File:Runcinated alternated cubic honeycomb verf.png|60px]]<br/>tapered [[triangular prism]] |- |J<sub>24</sub><br/>A<sub>20</sub><br/>W<sub>16</sub><br/>G<sub>21</sub><br/>h<sub>2,3</sub>δ<sub>4</sub><br/>O<sub>28</sub> |[[Runcicantic cubic honeycomb|Runcicantic cubic]] (gratoh)<br/>{{CDD|nodes_10ru|split2|node_1|4|node_1}} β {{CDD|node_h1|4|node|3|node_1|3|node_1}} |[[File:truncated hexahedron.png|30px]] (1)<br/>[[truncated cube|(3.8.8)]] | |[[File:Great rhombicuboctahedron.png|30px]](2)<br/>[[truncated cuboctahedron|(4.6.8)]] |[[File:truncated tetrahedron.png|30px]] (1)<br/>[[truncated tetrahedron|(3.6.6)]] |[[File:Cantitruncated alternated cubic honeycomb.png|75px]] |[[File:Cantitruncated alternated cubic tiling.png|75px]] |[[File:Runcitruncated alternate cubic honeycomb verf.png|60px]]<br/>Irregular [[tetrahedron]] |} {|class="wikitable" |+ <[4,3<sup>1,1</sup>]> uniform honeycombs, [[space group]] Pm{{overline|3}}m (221) |- !rowspan=2|Referenced<br/>indices !rowspan=2|Honeycomb name<br/>[[Coxeter diagram]]s<br/>{{CDD|nodeab_c1|split2|node_c2|4|node_c3}} β {{CDD|node_h0|4|node_c1|3|node_c2|4|node_c3}} !colspan=4|Cells by location<br/>(and count around each vertex) !rowspan=2|Solids<br/>(Partial) !rowspan=2|Frames<br/>(Perspective) !rowspan=2|[[vertex figure]] |- !(0,0')<br/>{{CDD|nodea|3a|nodea|4a|nodea}} !(1)<br/>{{CDD|nodea|2|nodeb|2|nodea}} !(3)<br/>{{CDD|nodea|3a|branch}} !Alt |- BGCOLOR="#e0f0e0" |J<sub>11,15</sub><br/>A<sub>1</sub><br/>W<sub>1</sub><br/>G<sub>22</sub><br/>δ<sub>4</sub><br/>O<sub>1</sub> |[[cubic honeycomb|Cubic]] (chon)<br/>{{CDD|nodes|split2|node|4|node_1}} β {{CDD|node_h0|4|node|3|node|3|node_1}} |[[File:hexahedron.png|30px]] (8)<br/>[[cube|(4.4.4)]] | | | |[[File:Bicolor cubic honeycomb.png|75px]] |[[File:Cubic tiling.png|75px]] |[[File:Cubic honeycomb verf.svg|60px]]<br/>[[octahedron]] |- BGCOLOR="#e0f0e0" |rowspan=2|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>δ<sub>4</sub><br/>O<sub>15</sub> |[[rectified cubic honeycomb|Rectified cubic]] (rich)<br/>{{CDD|nodes|split2|node_1|4|node}} β {{CDD|node_h0|4|node|3|node_1|4|node}} |[[File:cuboctahedron.png|30px]] (4)<br/>[[cuboctahedron|(3.4.3.4)]] | |[[File:Uniform polyhedron-33-t1.svg|30px]] (2)<br/>[[octahedron|(3.3.3.3)]] | |[[File:Rectified cubic honeycomb4.png|75px]] |rowspan=2|[[File:Rectified cubic tiling.png|75px]] |[[File:Rectified alternate cubic honeycomb verf.png|60px]]<br/>[[cuboid]] |- BGCOLOR="#e0f0e0" |[[rectified cubic honeycomb|Rectified cubic]] (rich)<br/>{{CDD|nodes_11|split2|node|4|node}} β {{CDD|node_h0|4|node_1|3|node|4|node}} |[[File:octahedron.png|30px]] (2)<br/>[[octahedron|(3.3.3.3)]] | |[[File:Uniform polyhedron-33-t02.svg|30px]] (4)<br/>[[cuboctahedron|(3.4.3.4)]] | |[[File:Rectified cubic honeycomb3.png|75px]] |[[File:Cantellated alternate cubic honeycomb verf.png|60px]]<br/>[[cuboid]] |- BGCOLOR="#e0f0e0" |J<sub>13</sub><br/>A<sub>14</sub><br/>W<sub>15</sub><br/>G<sub>8</sub><br/>t<sub>0,1</sub>δ<sub>4</sub><br/>O<sub>14</sub> |[[truncated cubic honeycomb|Truncated cubic]] (tich)<br/>{{CDD|nodes|split2|node_1|4|node_1}} β {{CDD|node_h0|4|node|3|node_1|4|node_1}} |[[File:truncated hexahedron.png|30px]] (4)<br/>[[truncated cube|(3.8.8)]] | |[[File:Uniform polyhedron-33-t1.svg|30px]] (1)<br/>[[octahedron|(3.3.3.3)]] | |[[File:Truncated cubic honeycomb2.png|75px]] |[[File:Truncated cubic tiling.png|75px]] |[[File:Bicantellated alternate cubic honeycomb verf.png|60px]]<br/>[[square pyramid]] |- BGCOLOR="#e0f0e0" |J<sub>14</sub><br/>A<sub>17</sub><br/>W<sub>12</sub><br/>G<sub>9</sub><br/>t<sub>0,2</sub>δ<sub>4</sub><br/>O<sub>17</sub> |[[cantellated cubic honeycomb|Cantellated cubic]] (srich)<br/>{{CDD|nodes_11|split2|node|4|node_1}} β {{CDD|node_h0|4|node_1|3|node|4|node_1}} |[[File:small rhombicuboctahedron.png|30px]] (2)<br/>[[rhombicuboctahedron|(3.4.4.4)]] |[[File:Uniform polyhedron 222-t012.png|30px]] (2)<br/>[[cube|(4.4.4)]] |[[File:Uniform polyhedron-33-t02.svg|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]] | |[[File:Cantellated cubic honeycomb.jpg|75px]] |[[File:Cantellated cubic tiling.png|75px]] |[[File:Runcicantellated alternate cubic honeycomb verf.png|60px]]<br/>obilique [[triangular prism]] |- BGCOLOR="#e0f0e0" |J<sub>16</sub><br/>A<sub>3</sub><br/>W<sub>2</sub><br/>G<sub>28</sub><br/>t<sub>0,2</sub>δ<sub>4</sub><br/>O<sub>16</sub> |[[bitruncated cubic honeycomb|Bitruncated cubic]] (batch)<br/>{{CDD|nodes_11|split2|node_1|4|node}} β {{CDD|node_h0|4|node_1|3|node_1|4|node}} |[[File:truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]] | |[[File:Uniform polyhedron-33-t012.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]] | |[[File:Bitruncated cubic honeycomb3.png|75px]] |[[File:Bitruncated cubic tiling.png|75px]] |[[File:Cantitruncated alternate cubic honeycomb verf.png|60px]]<br/>isosceles [[tetrahedron]] |- BGCOLOR="#e0f0e0" |J<sub>17</sub><br/>A<sub>18</sub><br/>W<sub>13</sub><br/>G<sub>25</sub><br/>t<sub>0,1,2</sub>δ<sub>4</sub><br/>O<sub>18</sub> |[[cantitruncated cubic honeycomb|Cantitruncated cubic]] (grich)<br/>{{CDD|nodes_11|split2|node_1|4|node_1}} β {{CDD|node_h0|4|node_1|3|node_1|4|node_1}} |[[File:Great rhombicuboctahedron.png|30px]] (2)<br/>[[truncated cuboctahedron|(4.6.8)]] |[[File:Uniform polyhedron 222-t012.png|30px]] (1)<br/>[[cube|(4.4.4)]] |[[File:Uniform polyhedron-33-t012.png|30px]](1)<br/>[[truncated octahedron|(4.6.6)]] | |[[File:Cantitruncated Cubic Honeycomb.svg|75px]] |[[File:Cantitruncated cubic tiling.png|75px]] |[[File:Omnitruncated alternated cubic honeycomb verf.png|60px]]<br/>irregular [[tetrahedron]] |- BGCOLOR="#d0f0f0" |J<sub>21,31,51</sub><br/>A<sub>2</sub><br/>W<sub>9</sub><br/>G<sub>1</sub><br/>hδ<sub>4</sub><br/>O<sub>21</sub> |[[Tetrahedral-octahedral honeycomb|Alternated cubic]] (octet)<br/>{{CDD|node_h1|4|node|split1|nodes}} β {{CDD|node_1|split1|nodes|split2|node}} |[[File:tetrahedron.png|30px]] (8)<br/>[[tetrahedron|(3.3.3)]] | | |[[File:Octahedron.png|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|60px]]<br/>[[cuboctahedron]] |- BGCOLOR="#d0f0f0" |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>δ<sub>4</sub><br/>O<sub>25</sub> |[[Cantic cubic honeycomb|Cantic cubic]] (tatoh)<br/>{{CDD|node_h1|4|node|split1|nodes_11}} β {{CDD|node_1|split1|nodes_11|split2|node}} |[[File:Truncated tetrahedron.png|30px]] (2)<br/>[[Truncated tetrahedron|(3.6.6)]] | |[[File:Cuboctahedron.png|30px]] (1)<br/>[[cuboctahedron|(3.4.3.4)]] |[[File:Truncated octahedron.png|30px]] (2)<br/>[[truncated octahedron|(4.6.6)]] ||[[File:Truncated Alternated Cubic Honeycomb.svg|75px]] |[[File:Truncated alternated cubic tiling.png|75px]] |[[File:Truncated alternated cubic honeycomb verf.png|60px]]<br/>rectangular pyramid |- BGCOLOR="#d0f0f0" |Nonuniform<sub>a</sub> |[[Bitruncated cubic honeycomb#Related polyhedra and honeycombs|Alternated bitruncated cubic]] (bisch)<br/>{{CDD|nodes_hh|split2|node_h|4|node}} β {{CDD|node_h0|4|node_h|3|node_h|4|node}} |[[File:Uniform polyhedron-43-h01.svg|30px]] (2)<br/>[[Regular icosahedron|(3.3.3.3.3)]] | |[[File:Uniform polyhedron-33-s012.svg|30px]] (2)<br/>[[Regular icosahedron|(3.3.3.3.3)]] |[[File:tetrahedron.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]] | | |[[File:Alternated bitruncated cubic honeycomb verf.png|60px]] |- BGCOLOR="#d0f0f0" |Nonuniform<sub>b</sub> |[[Alternated cantitruncated cubic honeycomb|Alternated cantitruncated cubic]] (serch)<br/>{{CDD|nodes_hh|split2|node_h|4|node_h}} β {{CDD|node_h0|4|node_h|3|node_h|4|node_h}} |[[File:snub hexahedron.png|30px]] (2)<br/>[[snub cube|(3.3.3.3.4)]] |[[File:tetrahedron.png|30px]] (1)<br/>[[tetrahedron|(3.3.3)]] |[[File:Uniform polyhedron-43-h01.svg|30px]] (1)<br/>[[Regular icosahedron|(3.3.3.3.3)]] |[[File:tetrahedron.png|30px]] (4)<br/>[[tetrahedron|(3.3.3)]] |[[File:Alternated cantitruncated cubic honeycomb.png|75px]] | |[[File:Alternated cantitruncated cubic honeycomb verf.png|60px]]<br/>Irr. [[tridiminished icosahedron]] |}
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