Dense packings of the Platonic and Archimedean solids

S. Torquato () and Y. Jiao
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S. Torquato: Department of Chemistry,
Y. Jiao: Princeton University, Princeton, New Jersey 08544, USA

Nature, 2009, vol. 460, issue 7257, 876-879

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.

Date: 2009
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DOI: 10.1038/nature08239

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