Dense packings of the Platonic and Archimedean solids.
Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline states of matter, granular media, heterogeneous materials and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory. 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.
Torquato
S
S
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA. torquato@princeton.edu
Jiao
Y
Y
PLoS Comput Biol. 2008 Aug 15;4(8):e1000152
18704170
Phys Rev A Gen Phys. 1985 Oct;32(4):2347-2351
9896349
Phys Rev Lett. 2004 Jun 25;92(25 Pt 1):255506
15245027
Biophys J. 2001 Aug;81(2):751-66
11463623
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 May;75(5 Pt 1):051304
17677051
Proc Natl Acad Sci U S A. 2006 Jul 11;103(28):10612-7
16818891
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Apr;79(4 Pt 1):041309
19518226
Proc Natl Acad Sci U S A. 2003 Mar 18;100(6):3173-8
12629206