Duration: 1:10:52

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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.