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 13.08.2009   Карта сайта     Language По-русски По-английски
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13.08.2009

Nature 460, 876-879 (13 August 2009) | doi:10.1038/nature08239; Received 29 April 2009; Accepted 24 June 2009



Dense packings of the Platonic and Archimedean solids


S. Torquato1,2,3,4,5 & Y. Jiao4




  1. Department of Chemistry,

  2. Princeton Center for Theoretical Science,

  3. Princeton Institute for the Science and Technology of Materials,

  4. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

  5. School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA


Correspondence to: S. Torquato1,2,3,4,5 Correspondence and requests for materials should be addressed to S.T. (Email: torquato@princeton.edu).





Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline states of matter1, 2, 3, 4, granular media3, 5, heterogeneous materials3 and biological systems6, 7, 8. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory9, 10, 11. Previous work has focused mainly on spherical particles—very little is known about dense polyhedral packings. Here we formulate the generation of dense packings of polyhedra as an optimization problem, using an adaptive fundamental cell subject to periodic boundary conditions (we term this the 'adaptive shrinking cell' scheme). Using a variety of multi-particle initial configurations, we find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space. The densities are 0.782ellipses, 0.947ellipses, 0.904... and 0.836..., respectively. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Combining our simulation results with derived rigorous upper bounds and theoretical arguments leads us to the 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.



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  • Chen Wev .  honorary member of ISSC science council

  • Harton Vladislav Vadim  honorary member of ISSC science council

  • Lichtenstain Alexandr Iosif  honorary member of ISSC science council

  • Novikov Dimirtii Leonid  honorary member of ISSC science council

  • Yakushev Mikhail Vasilii  honorary member of ISSC science council

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