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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity, and symmetries of geometric objects. Discret. Comput. Geom. 3, 237–256 (1988)
Andrews, L., Bernstein, H., Pelletier, G.: A perturbation stable cell comparison technique. Acta Crystallogr. A 36(2), 248–252 (1980)
Anosova, O., Kurlin, V.: Introduction to periodic geometry and topology. arXiv:2103.02749 (2021)
Bright, M., Anosova, O., Kurlin, V.: A proof of the invariant-based formula for the linking number and its asymptotic behaviour. In: Proceedings of NumGrid (2020)
Bouniaev, M., Dolbilin, N.: Regular and multi-regular t-bonded systems. J. Inf. Process. 25, 735–740 (2017)
Bright, M., Kurlin, V.: Encoding and topological computation on textile structures. Comput. Graph. 90, 51–61 (2020)
Dolbilin, N., Bouniaev, M.: Regular t-bonded systems in R\(^3\). Eur. J. Comb. 80, 89–101 (2019)
Dolbilin, N., Lagarias, J., Senechal, M.: Multiregular point systems. Discret. Comput. Geom. 20(4), 477–498 (1998)
Edelsbrunner, H., Heiss, T., Kurlin, V., Smith, P., Wintraecken, M.: The density fingerprint of a periodic point set. In: Proceedings of SoCG (2021)
Grünbaum, F., Moore, C.: The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets. Acta Crystallogr. A 51, 310–323 (1995)
Hahn, T., Shmueli, U., Arthur, J.: Intern. Tables crystallogr. 1, 750 (1983)
Hargreaves, C.J., Dyer, M.S., Gaultois, M.W., Kurlin, V.A., Rosseinsky, M.J.: The earth mover’s distance as a metric for the space of inorganic compositions. Chem. Mater. 32, 10610–10620 (2020)
Himanen, L., et al.: Dscribe: library of descriptors for machine learning in materials science. Comput. Phys. Commun. 247, 106949 (2020)
Mosca, M., Kurlin, V.: Voronoi-based similarity distances between arbitrary crystal lattices. Cryst. Res. Technol. 55(5), 1900197 (2020)
Patterson, A.: Ambiguities in the x-ray analysis of crystal structures. Phys. Rev. 65, 195 (1944)
Pozdnyakov, S., Willatt, M., Bartók, A., Ortner, C., Csányi, G., Ceriotti, M.: Incompleteness of atomic structure representations. Phys. Rev. Lett. 125, 166001 (2020). http://arxiv.org/abs/2001.11696
Pulido, A., et al.: Functional materials discovery using energy-structure-function maps. Nature 543, 657–664 (2017)
Widdowson, D., Mosca, M., Pulido, A., Kurlin, V., Cooper, A.: Average minimum distances of periodic point sets. arXiv:2009.02488 (2020)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Anosova, O., Kurlin, V. (2021). An Isometry Classification of Periodic Point Sets. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2021. Lecture Notes in Computer Science(), vol 12708. Springer, Cham. https://doi.org/10.1007/978-3-030-76657-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-76657-3_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-76656-6
Online ISBN: 978-3-030-76657-3
eBook Packages: Computer ScienceComputer Science (R0)
