  Motivated by the relation between particle shape and packing, we measure the
volume fraction $\phi$ occupied by the Platonic solids which are a class of
polyhedron with congruent sides, vertices and dihedral angles. Tetrahedron,
cube, octahedron, dodecahedron, and icosahedron shaped plastic dice were
fluidized or mechanically vibrated to find stable random loose packing
$\phi_{rlp} = 0.51, 0.54, 0.52, 0.51, 0.50$ and densest packing $\phi_{rcp} =
0.64, 0.67, 0.64, 0.63, 0.59$, respectively with standard deviation $\simeq \pm
0.01$. We find that $\phi$ obtained by all protocols peak at the cube, which is
the only Platonic solid that can tessellate space, and then monotonically
decrease with number of sides. This overall trend is similar but systematically
lower than the maximum $\phi$ reported for frictionless Platonic solids, and
below $\phi_{rlp}$ of spheres for the loose packings. Experiments with ceramic
tetrahedron were also conducted, and higher friction was observed to lead to
lower $\phi$.
