Particle Packing Problems for Fun and Profit

  • Salvatore Torquato | Princeton, Dept of Chemistry

Packing problems, such as how densely nonoverlapping particles fill d-dimensional Euclidean
space Rd are ancient and still provide fascinating challenges for scientists and mathematicians
[1,2]. Bernal has remarked that “heaps” (particle packings) were the first things that were
ever measured in the form of basketfuls of grain for the purpose of trading or of collection
of taxes. While maximally dense packings are intimately related to classical ground states
of matter, disordered sphere packings have been employed to model glassy states of matter.
There has been a resurgence of interest in maximally dense sphere packings in high-dimensional
Euclidean spaces [3,4], which is directly related to the optimal way of sending digital signals
over noisy channels.
I begin by first describing “order” maps to classify jammed sphere packings, which enables one
to view a host of packings with varying degrees of disorder as extremal structures. I discuss
work that provides the putative exponential improvement on a 100-year- old lower bound on
the maximal packing density due to Minkowski in Rd in the asymptotic limit d ? 8 [4].
Our study suggests that disordered (rather than ordered) sphere packings may be the densest
for sufficiently large d – a counterintuitive possibility. Finally, I describe recent work to find
and characterize dense packings of three-dimensional nonspherical shapes of various shapes,
including the Platonic and Archimedean solids [5]. We conjecture that the densest packings of
the Platonic and Archimedean solids with central symmetry are given by their corresponding
densest lattice packings. This is the analogue of Kepler’s sphere conjecture for these solids.

Speaker Details

Salvatore Torquato is Professor of Chemistry and the Princeton Institute for the Science and Technology of Materials at Princeton University. He is a Senior Faculty Fellow in the Princeton Center for Theoretical Science.
He also hold appointments in four departments at Princeton: Physics, Applied and Computational Mathematics, Chemical Engineering, and Mechanical & Aerospace Engineering. He is broadly interested in the fundamental microscopic understanding of the structure and bulk properties of condensed matter using statistical mechanics. His current work has been focused on self-assembly theory, particle packing problems, quasicrystals, optimal multifunctional material design and cancer modeling. He has published over 280 journal articles and a book entitled “Random Heterogeneous Materials.” He is the recipient of numerous awards/honors, including the American Physical Society 2009 Adler Lectureship Award for Materials Physics, Society for Industrial and Applied Mathematics 2007 Ralph E. Kleinman Prize and Society of Engineering Science 2004 William Prager Medal.

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