This honeycomb is one of seven unique uniform honeycombs[1] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A4 honeycombs
Pentagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
a1 [3[5]] (None)
i2 [[3[5]]] ×2  1, 2, 3,

 4, 5, 6

r10 [5[3[5]]] ×10  7

References

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  1. ^ mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks