1. Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA
2. Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil

Correspondence to: Hernán A. Makse^1,2 Correspondence and requests for materials should be addressed to H.A.M. (Email: hmakse@lev.ccny.cuny.edu).

The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured¹ by Kepler and the random geometries explored² by Bernal. Apart from its mathematical interest, the problem has practical relevance³ in a wide range of fields, from granular processing to fruit packing. There are currently numerous experiments showing that the loosest way to pack spheres (random loose packing) gives a density of 55 per cent^{4, 5, 6}. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of 64 per cent^{2, 4, 6}. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states⁷ in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. Our approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a density limit of 63.4 per cent. We construct a phase diagram that provides a unified view of the hard-sphere packing problem and illuminates various data, including the random-loose-packed state.

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