Maximally dense packings of two-dimensional convex and concave noncircular particles

Steven Atkinson, Yang Jiao, Salvatore Torquato

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38 Scopus citations


Dense packings of hard particles have important applications in many fields, including condensed matter physics, discrete geometry, and cell biology. In this paper, we employ a stochastic search implementation of the Torquato-Jiao adaptive-shrinking-cell (ASC) optimization scheme to find maximally dense particle packings in d-dimensional Euclidean space Rd. While the original implementation was designed to study spheres and convex polyhedra in d=3, our implementation focuses on d=2 and extends the algorithm to include both concave polygons and certain complex convex or concave nonpolygonal particle shapes. We verify the robustness of this packing protocol by successfully reproducing the known putative optimal packings of congruent copies of regular pentagons and octagons, then employ it to suggest dense packing arrangements of congruent copies of certain families of concave crosses, convex and concave curved triangles (incorporating shapes resembling the Mercedes-Benz logo), and "moonlike" shapes. Analytical constructions are determined subsequently to obtain the densest known packings of these particle shapes. For the examples considered, we find that the densest packings of both convex and concave particles with central symmetry are achieved by their corresponding optimal Bravais lattice packings; for particles lacking central symmetry, the densest packings obtained are nonlattice periodic packings, which are consistent with recently-proposed general organizing principles for hard particles. Moreover, we find that the densest known packings of certain curved triangles are periodic with a four-particle basis, and we find that the densest known periodic packings of certain moonlike shapes possess no inherent symmetries. Our work adds to the growing evidence that particle shape can be used as a tuning parameter to achieve a diversity of packing structures.

Original languageEnglish (US)
Article number031302
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number3
StatePublished - Sep 10 2012

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Statistics and Probability


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