Maximum and Minimum Stable Random Packings of Platonic Solids

Motivated by the relation between particle shape and packing, we measure the volume fraction $\varphi$ 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 ${\varphi }_{rlp}=0.51,0.54,0.52,0.51,0.50$ and densest packing $\phi_{rcp} = 0.64, 0.67, 0.64, 0.63, 0.59$,respectivelywithstandarddeviation$\simeq \pm 0.01$\mathrm{.}Wefindthat$\phi$ 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 $\varphi$ reported for frictionless Platonic solids, and below ${\varphi }_{rlp}$ of spheres for the loose packings. Experiments with ceramic tetrahedron were also conducted, and higher friction was observed to lead to lower $\varphi$.