Dense packings of the Platonic and Archimedean solids

Article metrics

  • An Erratum to this article was published on 08 October 2009
  • A Corrigendum to this article was published on 25 February 2010

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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).

References

  1. 1

    Bernal, J. D. in Liquids: Structure, Properties, Solid Interactions (eds Hughel, T. J.) 25–50 (Elsevier, 1965)

  2. 2

    Zallen, R. The Physics of Amorphous Solids (Wiley, 1983)

  3. 3

    Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, 2002)

  4. 4

    Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000)

  5. 5

    Edwards, S. F. in Granular Matter (eds Mehta, A.) 121–140 (Springer, 1994)

  6. 6

    Liang, J. & Dill, K. A. Are proteins well-packed? Biophys. J. 81, 751–766 (2001)

  7. 7

    Purohit, P. K., Kondev, J. & Phillips, R. Mechanics of DNA packaging in viruses. Proc. Natl Acad. Sci. USA 100, 3173–3178 (2003)

  8. 8

    Gevertz, J. L. & Torquato, S. A novel three-phase model of brain tissue microstructure. PLOS Comput. Biol. 4, e1000152 (2008)

  9. 9

    Conway, J. H. & Sloane, N. J. A. Sphere Packings, Lattices and Groups (Springer, 1998)

  10. 10

    Hales, T. C. A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)

  11. 11

    Cohn, H. & Elkies, N. New upper bounds on sphere packings. I. Ann. Math. 157, 689–714 (2003)

  12. 12

    Donev, A., Stillinger, F. H., Chaikin, P. M. & Torquato, S. Unusually dense crystal ellipsoid packings. Phys. Rev. Lett. 92, 255506 (2004)

  13. 13

    Conway, J. H. & Torquato, S. Packing, tiling and covering with tetrahedra. Proc. Natl Acad. Sci. USA 103, 10612–10617 (2006)

  14. 14

    Chen, E. R. A dense packing of regular tetrahedra. Discrete Comput. Geom. 40, 214–240 (2008)

  15. 15

    Jiao, Y., Stillinger, F. H. & Torquato, S. Optimal packings of superballs. Phys. Rev. E 79, 041309 (2009)

  16. 16

    Hoylman, D. J. The densest lattice packing of tetrahedra. Bull. Am. Math. Soc. 76, 135–137 (1970)

  17. 17

    Betke, U. & Henk, M. Densest lattice packings of 3-polytopes. Comput. Geom. 16, 157–186 (2000)

  18. 18

    Minkowski, H. Dichteste gitterförmige Lagerung kongruenter Körper. Nachr. Akad. Wiss. Göttingen Math. Phys. KI. II 311–355 (1904)

  19. 19

    Gardner, M. The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems Ch. 10 135 (Norton, 2001)

  20. 20

    Cromwell, P. R. Polyhedra (Cambridge Univ. Press, 1997)

  21. 21

    Jodrey, W. S. & Tory, E. M. Computer simulation of close random packing of equal spheres. Phys. Lett. A 32, 2347–2351 (1985)

  22. 22

    Rintoul, M. D. & Torquato, S. S. Hard-sphere statistics along the metastable amorphous branch. Phys. Rev. E 58, 532–537 (1998)

  23. 23

    Uche, O. U., Stillinger, F. H. & Torquato, S. Concerning maximal packing arrangements of binary disk mixtures. Physica A 342, 428–446 (2004)

  24. 24

    Donev, A., Torquato, S. & Stillinger, F. H. Neighbor list collision-driven molecular dynamics for nonspherical hard particles. I. Algorithmic details. J. Comput. Phys. 202, 737–764 (2005)

  25. 25

    Donev, A., Torquato, S. & Stillinger, F. H. Neighbor list collision-driven molecular dynamics for nonspherical hard particles. II. Applications to ellipses and ellipsoids. J. Comput. Phys. 202, 765–793 (2005)

  26. 26

    Golshtein, E. G. & Tretyakov, N. V. Modified Lagrangians and Monotone Maps in Optimization (Wiley, 1996)

  27. 27

    Torquato, S. & Stillinger, F. H. Multiplicity of generation, selection, and classification procedures for jammed hard-particle packings. J. Phys. Chem. B 105, 11849–11853 (2001)

  28. 28

    Donev, A., Connelly, R., Stillinger, F. H. & Torquato, S. Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids. Phys. Rev. E 75, 051304 (2007)

Download references

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.

Author information

Correspondence to S. Torquato.

Supplementary information

Supplementary Information

This file contains Supplementary Data and Supplementary Tables 1-2. (PDF 99 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.