An Isometry Classification of Periodic Point Sets

Abstract

We develop discrete geometry methods to resolve the data ambiguity challenge for periodic point sets to accelerate materials discovery. In any high-dimensional Euclidean space, a periodic point set is obtained from a finite set (motif) of points in a parallelepiped (unit cell) by periodic translations of the motif along basis vectors of the cell.

An important equivalence of periodic sets is a rigid motion or an isometry that preserves interpoint distances. This equivalence is motivated by solid crystals whose periodic structures are determined in a rigid form.

Crystals are still compared by descriptors that are either not isometry invariants or depend on manually chosen tolerances or cut-off parameters. All discrete invariants including symmetry groups can easily break down under atomic vibrations, which are always present in real crystals.

We introduce a complete isometry invariant for all periodic sets of points, which can additionally carry labels such as chemical elements. The main classification theorem says that any two periodic sets are isometric if and only if their proposed complete invariants (called isosets) are equal.

A potential equality between isosets can be checked by an algorithm, whose computational complexity is polynomial in the number of motif points. The key advantage of isosets is continuity under perturbations, which allows us to quantify similarities between any periodic point sets.

Supported by the EPSRC grant Application-driven Topological Data Analysis.