An algorithm for the arithmetic classification of multilattices

Abstract

A procedure for the construction and the classification of multilattices in arbitrary dimension is proposed. The algorithm allows to determine explicitly the location of the points of a multilattice given its space group, and to determine whether two multilattices are arithmetically equivalent. The algorithm is based on ideas from integer matrix theory, in particular the reduction to the Smith normal form. Among the applications of this procedure is a software package that allows the classification of complex crystalline structures and the determination of their space groups. Also, it can be used to determine the symmetry of regular systems of points in high dimension, with applications to the study of quasicrystals and sets of points with noncrystallographic symmetry in low dimension, such as viral capsid structures.