Trinomial triangle

This honeycomb is one of 20 uniform honycombs constructed by the ${\tilde {D}}_{5}$ 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:

Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,3,31,1]	Template:CDD	×1	Template:CDD
[[31,1,3,31,1]]	Template:CDD	×2	Template:CDD, Template:CDD
<[31,1,3,31,1]> = [31,1,3,3,4]	Template:CDD = Template:CDD	×2	Template:CDD, Template:CDD, Template:CDD, Template:CDD Template:CDD, Template:CDD, Template:CDD, Template:CDD
<<[31,1,3,31,1]>> = [4,3,3,3,4]	Template:CDD = Template:CDD	×4	Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD
[<<[31,1,3,1,1]>>] = [[4,3,3,3,4]]	Template:CDD = Template:CDD	×8	Template:CDD, Template:CDD, Template:CDD

References

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