The emerging area of Geometric Data Science studies real data objects under practical equivalences [1]. The key example is a point cloud under isometry or rigid motion. In a Euclidean space, if given points are ordered, their isometry class is uniquely determined by the matrix of pairwise distances. A naive extension to m unordered points requires exponentially many matrices depending on m! permutations. The case of m=3 points was settled 2000+ years ago due to the side-side-side theorem in school geometry. However, even for m=4 points in the plane, there are infinitely many 4-point clouds that are indistinguishable by 6 pairwise distances. The talk will describe a simple invariant that finished the case of 4 points [2] and a complete invariant with Lipschitz continuous metrics that resolved the exponential challenge in polynomial time in the number of points for any fixed Euclidean dimension [3]. [1] O.Anosova, V.Kurlin. Geometric Data Science book (arxiv:2512.05040), the latest version at https://kurlin.org/Geometric-Data-Science-book.pdf. [2] D.Widdowson, V.Kurlin. Resolving the data ambiguity for periodic crystals. NeurIPS 2022, v.35, p.24625-24638. Extended in SIAM J Applied Mathematics, v.86 (3), p.898-918 (2026). [3] D.Widdowson, V.Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives, CVPR 2023. Extended in arxiv:2303.14161 and MATCH 2026.