This honeycomb is one of 20 uniform honeycombs constructed by the <math>{\tilde{D}}_5</math> Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
| D5 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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| [31,1,3,31,1]
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File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
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<math>{\tilde{D}}_5</math>
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File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png
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<[31,1,3,31,1]> ↔ [31,1,3,3,4]
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File:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.pngFile:CDel split1.pngFile:CDel nodeab c5.png ↔ File:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c4.pngFile:CDel 3.pngFile:CDel node c5.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_5</math>×21 = <math>{\tilde{B}}_5</math>
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File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png
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| [[31,1,3,31,1]]
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File:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.png
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<math>{\tilde{D}}_5</math>×22
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File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png, File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png
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<2[31,1,3,31,1]> ↔ [4,3,3,3,4]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c3.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c4.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 3.pngFile:CDel node c4.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_5</math>×41 = <math>{\tilde{C}}_5</math>
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File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png
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[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]
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File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c2.pngFile:CDel 3.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 c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node.png
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<math>{\tilde{D}}_5</math>×8 = <math>{\tilde{C}}_5</math>×2
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File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png, File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png
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References