A method for dense packing discovery

TL;DR

This paper introduces a numerical method based on the divide and concur framework for discovering dense particle packings, successfully reproducing and improving known packings in multiple dimensions and for various particle shapes.

Contribution

The paper presents a novel general search method for dense packings that integrates unit cell parameters, enabling the discovery and verification of dense packings across different dimensions and particle types.

Findings

Reproduced densest known lattice sphere packings up to 14 dimensions.

Reproduced best known lattice kissing arrangements up to 11 dimensions.

Discovered a new dense packing of four-dimensional simplices with density approximately 0.5845.

Abstract

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.