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arXiv · 2506.08934

Piecewise-linear embeddings of the space of 3D lattices into $\RR^{13}$ for high-throughput handling of lattice parameters

Ryoko Oishi-Tomiyasu
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Abstract

We present two methods to continuously and piecewise-linearly parametrize rank-3 lattices by vectors of $\RR^{13}$, which provides an efficient way to judge if two sets of parameters provide nearly identical lattices within their margins of errors. Such a parametrization can be used to speed up scientific computing involving periodic structures in $\RR^3$ such as crystal structures, which includes database querying, detection of duplicate entries, and structure generation via deep learning techniques. One gives a novel application of Conway's vonorms and conorms, and another is achieved through a natural extension of Ry{\u s}hkov's $C$-type to the setting modulo $3$. Voronoi vectors modulo 3 obtained in the latter approach provide an algorithm for enumerating of all potential isometries under perturbations of lattice parameters.

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Ryoko Oishi-Tomiyasu. 2025-06-10. Piecewise-linear embeddings of the space of 3D lattices into $\RR^{13}$ for high-throughput handling of lattice parameters. https://arxiv.org/abs/2506.08934

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