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One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n :
Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]]
For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to
(d - 2)!!… Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing r i Exp[ n θ i ] , and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions.