  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.
