Lambda g conjecture

This honeycomb is one of ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ 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), [3^1,1,3,4,1⁺] (index 2) are all isomorphic to [3^1,1,1,1]. The ten permutations are listed with its highest extended symmetry relation:

Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,1,1]	Template:CDD	×1	(none)
<[31,1,1,1]> = [31,1,3,4]	Template:CDD = Template:CDD	×2	(none)
<<[1,131,1]>> = [4,3,3,4]	Template:CDD = Template:CDD	×4	Template:CDD 1, Template:CDD 2
[3[3,31,1,1]] = [3,4,3,3]	Template:CDD = Template:CDD	×6	Template:CDD3, Template:CDD 4, Template:CDD 5, Template:CDD 6
[<<[1,131,1]>>] = [[4,3,3,4]]	Template:CDD = Template:CDD	×8	Template:CDD 7, Template:CDD 8, Template:CDD 9, Template:CDD 10
[(3,3)[31,1,1,1]] = [3,3,4,3]	Template:CDD = Template:CDD	×24

References

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