  We formulate the problem of generating dense packings of nonoverlapping,
non-tiling 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 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 non-tiling 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. 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. In addition, we conjecture that the
optimal packing of any convex, congruent polyhedron without central symmetry
generally is not a lattice packing.
