Random Close Packing

The concept of "random close packing" was shown by Torquato et al. (2000) to be mathematically ill-defined idea that is better replaced by the notion of "maximally random jammed."

Random close packing of circles in two dimensions has a theoretical packing density of 0.886441 (Zaccone 2022).

Random close packing of spheres in three dimensions gives a packing density of only eta approx 0.64 (Bernal and Mason 1960, Jaeger and Nagel 1992, Zaccone 2022), significantly smaller than the optimal packing density for cubic or hexagonal close packing of 0.74048. Zaccone (2022) give an exact packing density of


using Percus-Yevick theory, or


using a "very accurate" Carnahan-Starling expression.

Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

See also

Cubic Close Packing, Ellipsoid Packing, Hexagonal Close Packing, Sphere Packing

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--. "What Is Random Packing." Nature 239, 488-489, 1972.Bernal, J. D. and Mason, J. "Packing of Spheres: Co-Ordination of Randomly Packed Spheres." Nature 188, 910-911, 1960.Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; and Chaikin, P. M. "Improving the Density of Jammed Disordered Packings using Ellipsoids." Science, 303, 990-993, 2004.Jaeger, H. M. and Nagel, S. R. "Physics of Granular States." Science 255, 1524, 1992.Reuters, Inc. "M&M's Obsession Leads to Physics Discovery.", S.; Truskett, T. M.; and Debenedetti, P. G. "Is Random Close Packing of Spheres Well Defined?" Phys. Lev. Lett. 84, 2064-2067, 2000.Zaccone, A. "Explicit Analytical Solution for Random Close Packing in d=2 and d=3." Phys. Rev. Lett. 128, 028002, pp. 1-5, 2022.

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Random Close Packing

Cite this as:

Weisstein, Eric W. "Random Close Packing." From MathWorld--A Wolfram Web Resource.

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