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This paper develops a new continuous approach to a similarity betweenperiodic lattices of ideal crystals. Quantifying a similarity between crystalstructures is needed to substantially speed up the Crystal StructurePrediction, because the prediction of many target properties of crystalstructures is computationally slow and is essentially repeated for many nearlyidentical simulated structures. The proposed distances between arbitraryperiodic lattices of crystal structures are invariant under all rigid motions,satisfy the metric axioms and continuity under atomic perturbations. The aboveproperties make these distances ideal tools for clustering and visualizinglarge datasets of crystal structures. All the conclusions are rigorously provedand justified by experiments on real and simulated crystal structures reportedin the Nature 2017 paper "Functional materials discovery usingenergy-structure-function maps".

Voronoi-based similarity distances between arbitrary crystal lattices