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Dense packings of the Platonic and Archimedean solids

Nature volume 460, pages 876–879 (2009)Cite this article

A Corrigendum to this article was published on 25 February 2010

An Erratum to this article was published on 08 October 2009

Abstract

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 0.782…, 0.947…, 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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Figure 1: The five Platonic solids and the 13 Archimedean solids.
Figure 2: Sequential changes of a four-particle packing configuration according to the design variables in the ASC algorithm.
Figure 3: Portions of the densest packing of tetrahedra obtained from our simulations, and the optimal lattice packings of the icosahedra, dodecahedra and octahedra to which our simulations converge.
Figure 4: Comparison of the densest known lattice packings (blue circles) of the Platonic and Archimedean solids 16,17,18 to the corresponding upper bounds (red squares) obtained from bound (3).

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Acknowledgements

We are grateful to H. Cohn and J. Conway for comments on our manuscript. S.T. thanks the Institute for Advanced Study for its hospitality during his stay there. This work was supported by the National Science Foundation under award numbers DMS-0804431 and DMR-0820341. The figures showing the polyhedra were generated using the AntiPrism package developed by A. Rossiter.

Author Contributions S.T. devised the algorithm and upper bounds, performed theoretical analysis, and wrote the paper. Y.J. implemented the algorithm, performed theoretical analysis, commented on the manuscript and created all of the figures.

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Authors and Affiliations

  1. Department of Chemistry,,

    S. Torquato

  2. Princeton Center for Theoretical Science,,

    S. Torquato

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

    S. Torquato

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

    S. Torquato & Y. Jiao

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

    S. Torquato

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  1. S. Torquato
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  2. Y. Jiao
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Correspondence to S. Torquato.

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Torquato, S., Jiao, Y. Dense packings of the Platonic and Archimedean solids. Nature 460, 876–879 (2009). https://doi.org/10.1038/nature08239

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