Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2009-08-27
Nature, vol. 460, 876-879 (2009)
Physics
Condensed Matter
Statistical Mechanics
18 pages, 4 figures. In the published version of this paper in Nature (described below), "P6" should be "A1" in Figure 1
Scientific paper
Dense packings have served as useful models of the structure of liquid, glassy and crystal 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 long-standing interest in discrete geometry and number theory. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong 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 analog of Kepler's sphere conjecture for these solids.
Jiao Yangxiu
Torquato Salvatore
No associations
LandOfFree
Dense Packings of the Platonic and Archimedean Solids does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Dense Packings of the Platonic and Archimedean Solids, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dense Packings of the Platonic and Archimedean Solids will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-192839