Optimal packing problems, such as how densely hard particles can fill a volume, have fascinated people since the dawn of civilization and the fascination persists in mathematics and science. Resurgent interest comes from the recent proof of the Kepler conjecture that the face- centered-cubic lattice provides the highest possible packing fraction *y = p */√18 » 0.74 of congruent spheres in three dimensions [1]. Random particle packings have been studied by biologists, materials scientists, engineers, chemists and physicists to understand the structure and bulk properties of living cells, liquids, composites, granular media, glasses and amorphous solids, to mention but a few examples [2].

Two recent concepts, jamming and order metrics, facilitate the characterization of optimal packings. The former is related to the mechanical stability of the packing [3,4,5] and the latter describes the degree of randomness of the packing via a scalar order metric *y* [2,3] such that *y* = 1 for perfect order and *y* = 0 for perfect disorder. Besides the FCC (*y* = *p* /√18, *y* = 1) in the *y−f* plane for sphere packings, I will discuss other interesting optimal points, including the maximally random jammed (MRJ) state, which replaces the venerable but ill-defined random close packed (RCP) state, as well as the jammed state with the smallest density. Recently, we have shown experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely; up to *f* = 0.68 - 0.71 for spheroids with an aspect ratio close to that of *M&M*'s Candies, and even approach *y* » 0.74 for general ellipsoids [6]. This has interesting implications for the existence of a thermodynamically stable glass. Moreover, we have discovered a family of crystal (periodic) ellipsoid packings with the highest known density [7].

1. T. C. Hales, “An Overview of the Kepler Conjecture,” http://xxx.lanl.gov/math.MG/9811071 (1998).

2. S. Torquato, “Random Heterogeneous Materials: Microstructure and Macroscopic Properties,” (Springer-Verlag, New York, 2002).

3. S. Torquato, T. M. Truskett and P. G. Debenedetti, “Is Random Close Packing of Spheres Well Defined?,” Phys. Rev. Lett. **84**, 2064 (2000).

4. S. Torquato and F. H. Stillinger, “Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings,” J. Phys. Chem. **105**, 11849 (2001).

5. A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, “Jamming in Hard Sphere and Disk Packings,” J. Appl. Phys. **95**, 989 (2004).

6. A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato, and P. M. Chaikin, “Improving the Density of Jammed Disordered Packings using Ellipsoids,” Science **33**, 690 (2004).

7. A. Donev, F. H. Stillinger, P. M. Chaikin, and S. Torquato, “Unusually Dense Crystal Ellipsoid Packings,” Phys. Rev. Lett. **92**, 255506 (2004).

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