Pointwise distance distributions of periodic sets

6 The fundamental model of a periodic structure is a periodic set of points considered up to rigid 7 motion or isometry in Euclidean space. The recent work by Edelsbrunner et al defined the new 8 isometry invariants (density functions), which are continuous under perturbations of points and 9 complete for generic sets in dimension 3. This work introduces much faster invariants called higher 10 order Pointwise Distance Distributions (PDD). The new PDD invariants are simpler represented 11 by numerical matrices and are also continuous under perturbations important for applications. 12 Completeness of PDD invariants is proved for distance-generic sets in any dimension, which was also 13 confirmed by distinguishing all 229K known molecular organic structures from the world’s largest 14 Cambridge Structural Database. This huge experiment took only seven hours on a modest desktop 15 due to the proposed algorithm with a near linear or small polynomial complexity in terms of key 16 input sizes. Most importantly, the above completeness allows one to build a common map of all 17 periodic structures, which are continuously parameterized by PDD and explicitly reconstructible 18 from PDD. Appendices include first tree-based maps for several thousands of real structures. 19 2012 ACM Subject Classification Theory of computation → Computational geometry 20