Pointwise distance distributions of periodic point sets

Abstract

The fundamental model of a periodic structure is a periodic set of points considered up to rigid motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new isometry invariants (density functions), which are continuous under perturbations of points and complete for generic sets in dimension 3. This work introduces much faster invariants called higher order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented by numerical matrices and are also continuous under perturbations important for applications.Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also confirmed by distinguishing all 229K known molecular organic structures from the world's largest Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop due to the proposed algorithm with a near linear or small polynomial complexity in terms of key input sizes. Most importantly, the above completeness allows one to build a common map of all periodic structures, which are continuously parameterized by PDD and explicitly reconstructible from PDD. Appendices include first tree-based maps for several thousands of real structures.

Exactly computable and continuous metrics on isometry classes of finite and 1-periodic sequences

“…3 emerged as first examples with identical infinite distributions of distances (or diffraction patterns). They are distinguished by recent Pointwise Distance Distributions [34] but not by the simpler Average Minimum Distances [35]. However, the latter invariants distinguish the even more interesting periodic sequences S 15 = {0, 1, 3, 4, 5, 7, 9, 10, 12} + 15Z and Q 15 = {0, 1, 3, 4, 6, 8, 9, 12, 14} + 15Z.…”

“…The recent developments in Periodic Geometry include continuous maps of Lattice Isometry Spaces in dimension two [23,8] and three [21,9], and applications to materials science [31,37]. The latest ultra-fast and generically complete Pointwise Distance Distributions [34] justified the Crystal Isometry Principle (CRISP) saying that all real periodic crystals live in a common space of isometry classes of periodic point sets continuously parameterised by their complete invariants such as isosets [3]. Any centered α-cluster representing a point p in a periodic sequence S can be considered as a list of vectors or signed distances from p to its neighbors within S up to the radius α.…”

A complete isometry classification of 3-dimensional lattices

“…The more recent Pointwise Distance Distributions [26] are continuous, complete for distance-generic crystals and helped establish the Crystal Isometry Principle saying that all real periodic crystals can be distinguished up to isometry by their geometric structures of atomic centres without chemical data. Hence all periodic crystals live in the common Crystal Isometry Space (CRISP), which can be projected to the Lattice Isometry Space LIS(R 3 ) parameterised in Problem 1.1.…”

Density functions of periodic sequences

“…For rational-valued periodic sequences, [5,Theorem 4] proved that r-th order invariants (combinations of r-factor products) up to r = 6 are enough to distinguish such sequences up to a shift (a rigid motion of R without reflections). The AMD invariant was extended to a Pointwise Distance Distribution (PDD), whose generic completeness [10,Theorem 11] was proved in any dimension n ≥ 1, but there are finite sets in R 3 with the same PDD [8, Fig. S4].…”

“…For S = {0, The recent developments in Periodic Geometry include continuous maps of Lattice Isometry Spaces in dimension two [7,2] and three [6,3], Pointwise Distance Distributions [10], and applications to materials science [9,12].…”