  A mechanically-based structural optimization method is utilized to explore
the phenomena of jamming for assemblies of frictionless Platonic solids.
Systems of these regular convex polyhedra exhibit mechanically stable phases
with density substantially less than optimal for a given shape, revealing that
thermal motion is necessary to access high density phases. We confirm that the
large system jamming threshold of $0.623 \pm 0.003$ for tetrahedra is
consistent with experiments on tetrahedral dice. Also, the extremely
short-ranged translational correlations of packed tetrahedra observed in
experiments are confirmed here, in contrast with those of thermally simulated
glasses. Though highly ordered phases are observed to form for small numbers of
cubes and dodecahedra, the short correlation length scale suppresses ordering
in large systems, resulting in packings that are mechanically consistent with
`orientationally disordered' contacts (point-face and edge-edge contacts). Mild
nematic ordering is observed for large systems of cubes, whereas angular
correlations for the remaining shapes are ultra short-ranged. Power-law scaling
exponents for energy with respect to distance from the jamming threshold
exhibit a clear dependence on the `highest order' percolating contact topology.
These nominal exponents are 6, 4, and 2 for configurations having percolating
point-face (or edge-edge), edge-face, and face-face contacts, respectively.
These Platonic solids exhibit hypostatic behavior, with average jamming contact
number between the isostatic value for spheres and that of asymmetric
particles. These shapes violate the isostatic conjecture, displaying contact
number that decreases monotonically with sphericity. The common symmetry of
dual polyhedra results in local translational structural similarity. (Note
abstract abridged due to length constraint)
