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Dense packings of the Platonic and Archimedean solids

Author

Listed:
  • S. Torquato

    (Department of Chemistry,
    Princeton Center for Theoretical Science,
    Princeton Institute for the Science and Technology of Materials,
    Princeton University, Princeton, New Jersey 08544, USA)

  • Y. Jiao

    (Princeton University, Princeton, New Jersey 08544, USA)

Abstract

Platonic affairs: packing the Platonic and Archimedean solids Models based on knowledge of the geometry of dense particle packing help explain the structure of many systems, including liquids, glasses, crystals, granular media and biological systems. Most previous work in this area has focused on spherical particles, but even for this idealized shape the problem is notoriously difficult — Kepler's conjecture on the densest packing of spheres was proved only in 2005. Little is known about the densest arrangements of the 18 classic geometric shapes, the Platonic and Archimedean solids, though they have been known since the time of the Ancient Greeks. Salvatore Torquato and Yang Jiao now report the densest known packings of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and 13 Archimedean polyhedra. The symmetries of the solids are crucial in determining their fundamental packing arrangements, and the densest packings of Platonic and Archimedean solids with central symmetry are conjectured to be given by their corresponding densest (Bravais) lattice packings.

Suggested Citation

  • S. Torquato & Y. Jiao, 2009. "Dense packings of the Platonic and Archimedean solids," Nature, Nature, vol. 460(7257), pages 876-879, August.
    • Handle: RePEc:nat:nature:v:460:y:2009:i:7257:d:10.1038_nature08239
    DOI: 10.1038/nature08239
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    Cited by:

    1. Liu, Lufeng & Lu, Peng & Meng, Lingyi & Jin, Weiwei & Li, Shuixiang, 2016. "Order metrics and order maps of octahedron packings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 870-882.
    2. Hongmei Liu & Shenglu Lu & Yingbo Zhang & Hui Chen & Yungui Chen & Ma Qian, 2022. "Migration of solidification grain boundaries and prediction," Nature Communications, Nature, vol. 13(1), pages 1-12, December.
    3. Xiaobin Dai & Xuanyu Zhang & Lijuan Gao & Ziyang Xu & Li-Tang Yan, 2022. "Topology mediates transport of nanoparticles in macromolecular networks," Nature Communications, Nature, vol. 13(1), pages 1-8, December.
    4. Romanova, T. & Bennell, J. & Stoyan, Y. & Pankratov, A., 2018. "Packing of concave polyhedra with continuous rotations using nonlinear optimisation," European Journal of Operational Research, Elsevier, vol. 268(1), pages 37-53.

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