Continuous Invariant-Based Maps of the Cambridge Structural Database.

The Cambridge Structural Database (CSD) played a key role in the recently established crystal isometry principle (CRISP). The CRISP says that any real periodic crystal is uniquely determined as a rigid structure by the geometry of its atomic centers without atomic types. Ignoring atomic types allows us to study all periodic crystals in a common space whose continuous nature is justified by the continuity of real-valued coordinates of atoms.

The importance of definitions in crystallography.

This paper was motivated by the articles `Same or different - that is the question' in CrystEngComm (July 2020) and `Change to the definition of a crystal' in the IUCr Newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3D space.

Accelerating material property prediction using generically complete isometry invariants.

Periodic material or crystal property prediction using machine learning has grown popular in recent years as it provides a computationally efficient replacement for classical simulation methods. A crucial first step for any of these algorithms is the representation used for a periodic crystal. While similar objects like molecules and proteins have a finite number of atoms and their representation can be built based upon a finite point cloud interpretation, periodic crystals are unbounded in size, making their representation more challenging.

Continuous chiral distances for two-dimensional lattices.

Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of two-dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographic-style coordinates. The four non-oblique Bravais classes of two-dimensional lattices form low-dimensional singular subspaces in the full continuous space.

Geographic style maps for two-dimensional lattices.

This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere.

Densest plane group packings of regular polygons.

Packings of regular convex polygons (n-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries.

One class classification as a practical approach for accelerating π-π co-crystal discovery.

The implementation of machine learning models has brought major changes in the decision-making process for materials design. One matter of concern for the data-driven approaches is the lack of negative data from unsuccessful synthetic attempts, which might generate inherently imbalanced datasets. We propose the application of the one-class classification methodology as an effective tool for tackling these limitations on the materials design problems.