Dense packings of polyhedra: Platonic and Archimedean solids

Abstract

Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. 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 nonoverlapping, nontiling 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 optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823..., 0.836..., 0.904..., and 0.947..., respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the "asphericity" (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be the analog of Kepler's sphere conjecture for these solids.The truncated tetrahedron is the only nonchiral Archimedean solid that is not centrally symmetric [corrected], the densest known packing of which is a non-lattice packing with density at least as high as 23/24=0.958 333... . We discuss the validity of our conjecture to packings of superballs, prisms, and antiprisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.