Measures of order for imperfect two-dimensional patterns

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures.  We also study imperfect hexagonal, square, and rhombic arrangements of nanodots produced by numerical simulations of pattern-forming partial differential equations and apply the measures to study packing on surfaces. 



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