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:
| D4 honeycombs
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Extended symmetry
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Extended diagram
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Extended group
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Honeycombs
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| [31,1,1,1]
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File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
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<math>{\tilde{D}}_4</math>
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(none)
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<[31,1,1,1]> ↔ [31,1,3,4]
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File:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c4.png ↔ File:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_4</math>×2 = <math>{\tilde{B}}_4</math>
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(none)
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<2[1,131,1]> ↔ [4,3,3,4]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c2.png ↔ File:CDel node.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_4</math>×4 = <math>{\tilde{C}}_4</math>
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File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png 1, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png 2
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[3[3,31,1,1]] ↔ [3,3,4,3]
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File:CDel node c3.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel splitsplit1.pngFile:CDel branch3 c1.pngFile:CDel node c1.png ↔ File:CDel node c3.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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<math>{\tilde{D}}_4</math>×6 = <math>{\tilde{F}}_4</math>
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File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel splitsplit1.pngFile:CDel branch3.pngFile:CDel node.png3, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel splitsplit1.pngFile:CDel branch3.pngFile:CDel node.png 4, File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel splitsplit1.pngFile:CDel branch3 11.pngFile:CDel node 1.png 5, File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel splitsplit1.pngFile:CDel branch3 11.pngFile:CDel node 1.png 6
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[4[1,131,1]] ↔ [[4,3,3,4]]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c1.png ↔ File:CDel node.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_4</math>×8 = <math>{\tilde{C}}_4</math>×2
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File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png 7, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png 8, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png 9
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[(3,3)[31,1,1,1]] ↔ [3,4,3,3]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c1.png ↔ File:CDel node c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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<math>{\tilde{D}}_4</math>×24 = <math>{\tilde{F}}_4</math>
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[(3,3)[31,1,1,1]]+ ↔ [3+,4,3,3]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c1.png ↔ File:CDel node c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
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½<math>{\tilde{D}}_4</math>×24 = ½<math>{\tilde{F}}_4</math>
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File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel split1.pngFile:CDel nodes hh.png 10
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References