Difference between revisions of "Template:D4 honeycombs"
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|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |<[3<sup>1,1,1,1</sup>]><BR> | + | |<[3<sup>1,1,1,1</sup>]><BR>↔ [3<sup>1,1</sup>,3,4] |
| − | |{{CDD|nodeab_c1-2|split2|node_c3|split1|nodeab_c4}}<BR> | + | |{{CDD|nodeab_c1-2|split2|node_c3|split1|nodeab_c4}}<BR>↔ {{CDD|nodeab_c1-2|split2|node_c3|3|node_c4|4|node}} |
| <math>{\tilde{D}}_4</math>×2 = <math>{\tilde{B}}_4</math> | | <math>{\tilde{D}}_4</math>×2 = <math>{\tilde{B}}_4</math> | ||
|(none) | |(none) | ||
|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |<2[<sup>1,1</sup>3<sup>1,1</sup>]><BR> | + | |<2[<sup>1,1</sup>3<sup>1,1</sup>]><BR>↔ [4,3,3,4] |
| − | |{{CDD|nodeab_c1|split2|node_c3|split1|nodeab_c2}}<BR> | + | |{{CDD|nodeab_c1|split2|node_c3|split1|nodeab_c2}}<BR>↔ {{CDD|node|4|node_c1|3|node_c3|3|node_c2|4|node}} |
| <math>{\tilde{D}}_4</math>×4 = <math>{\tilde{C}}_4</math> | | <math>{\tilde{D}}_4</math>×4 = <math>{\tilde{C}}_4</math> | ||
|{{CDD|nodes_11|split2|node|split1|nodes}} [[Rectified tesseractic honeycomb|<sub>1</sub>]], {{CDD|nodes_11|split2|node_1|split1|nodes}} [[Bitruncated tesseractic honeycomb|<sub>2</sub>]] | |{{CDD|nodes_11|split2|node|split1|nodes}} [[Rectified tesseractic honeycomb|<sub>1</sub>]], {{CDD|nodes_11|split2|node_1|split1|nodes}} [[Bitruncated tesseractic honeycomb|<sub>2</sub>]] | ||
|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |[3[3,3<sup>1,1,1</sup>]]<BR> | + | |[3[3,3<sup>1,1,1</sup>]]<BR>↔ [3,3,4,3] |
| − | |{{CDD|node_c3|3|node_c2|splitsplit1|branch3_c1|node_c1}}<BR> | + | |{{CDD|node_c3|3|node_c2|splitsplit1|branch3_c1|node_c1}}<BR>↔ {{CDD|node_c3|3|node_c2|3|node_c1|4|node|3|node}} |
| <math>{\tilde{D}}_4</math>×6 = <math>{\tilde{F}}_4</math> | | <math>{\tilde{D}}_4</math>×6 = <math>{\tilde{F}}_4</math> | ||
|{{CDD|node_1|3|node|splitsplit1|branch3|node}}[[16-cell honeycomb|<sub>3</sub>]], {{CDD|node_1|3|node_1|splitsplit1|branch3|node}} [[Truncated 16-cell honeycomb|<sub>4</sub>]], {{CDD|node|3|node|splitsplit1|branch3_11|node_1}} [[Birectified 16-cell honeycomb|<sub>5</sub>]], {{CDD|node|3|node_1|splitsplit1|branch3_11|node_1}} [[Bitruncated 16-cell honeycomb|<sub>6</sub>]] | |{{CDD|node_1|3|node|splitsplit1|branch3|node}}[[16-cell honeycomb|<sub>3</sub>]], {{CDD|node_1|3|node_1|splitsplit1|branch3|node}} [[Truncated 16-cell honeycomb|<sub>4</sub>]], {{CDD|node|3|node|splitsplit1|branch3_11|node_1}} [[Birectified 16-cell honeycomb|<sub>5</sub>]], {{CDD|node|3|node_1|splitsplit1|branch3_11|node_1}} [[Bitruncated 16-cell honeycomb|<sub>6</sub>]] | ||
|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |[4[<sup>1,1</sup>3<sup>1,1</sup>]]<BR> | + | |[4[<sup>1,1</sup>3<sup>1,1</sup>]]<BR>↔ [<span/>[4,3,3,4]] |
| − | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR> | + | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR>↔ {{CDD|node|4|node_c1|3|node_c2|3|node_c1|4|node}} |
| <math>{\tilde{D}}_4</math>×8 = <math>{\tilde{C}}_4</math>×2 | | <math>{\tilde{D}}_4</math>×8 = <math>{\tilde{C}}_4</math>×2 | ||
|rowspan=2|{{CDD|nodes|split2|node_1|split1|nodes}} [[24-cell honeycomb|<sub>7</sub>]], {{CDD|nodes_11|split2|node|split1|nodes_11}} [[Rectified 24-cell honeycomb|<sub>8</sub>]], {{CDD|nodes_11|split2|node_1|split1|nodes_11}} [[Truncated 24-cell honeycomb|<sub>9</sub>]] | |rowspan=2|{{CDD|nodes|split2|node_1|split1|nodes}} [[24-cell honeycomb|<sub>7</sub>]], {{CDD|nodes_11|split2|node|split1|nodes_11}} [[Rectified 24-cell honeycomb|<sub>8</sub>]], {{CDD|nodes_11|split2|node_1|split1|nodes_11}} [[Truncated 24-cell honeycomb|<sub>9</sub>]] | ||
|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |[(3,3)[3<sup>1,1,1,1</sup>]]<BR> | + | |[(3,3)[3<sup>1,1,1,1</sup>]]<BR>↔ [3,4,3,3] |
| − | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR> | + | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR>↔ {{CDD|node_c2|3|node_c1|4|node|3|node|3|node}} |
| <math>{\tilde{D}}_4</math>×24 = <math>{\tilde{F}}_4</math> | | <math>{\tilde{D}}_4</math>×24 = <math>{\tilde{F}}_4</math> | ||
|- style="text-align:center;" | |- style="text-align:center;" | ||
| − | |[(3,3)[3<sup>1,1,1,1</sup>]]<sup>+</sup><BR> | + | |[(3,3)[3<sup>1,1,1,1</sup>]]<sup>+</sup><BR>↔ [3<sup>+</sup>,4,3,3] |
| − | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR> | + | |{{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR>↔ {{CDD|node_c2|3|node_c1|4|node|3|node|3|node}} |
| ½<math>{\tilde{D}}_4</math>×24 = ½<math>{\tilde{F}}_4</math> | | ½<math>{\tilde{D}}_4</math>×24 = ½<math>{\tilde{F}}_4</math> | ||
|{{CDD|nodes_hh|split2|node_h|split1|nodes_hh}} [[snub 24-cell honeycomb|<sub>10</sub>]] | |{{CDD|nodes_hh|split2|node_h|split1|nodes_hh}} [[snub 24-cell honeycomb|<sub>10</sub>]] | ||
Revision as of 23:54, 19 February 2016
There are ten uniform honeycombs constructed by the <math>{\tilde{D}}_4</math> Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].
The ten permutations are listed with its highest extended symmetry relation: