Abstract: 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 discuss applications of the measures.