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A space for lattice representation and clustering
Author(s) -
Andrews Lawrence C.,
Bernstein Herbert J.,
Sauter Nicholas K.
Publication year - 2019
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273319002729
Subject(s) - bravais lattice , cluster analysis , lattice (music) , metric space , mathematics , delaunay triangulation , combinatorics , computer science , algorithm , discrete mathematics , physics , crystal structure , crystallography , statistics , chemistry , acoustics
Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation‐stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree‐based nearest‐neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation‐stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six‐dimensional real vectors or equivalently as three‐dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)