Maximum and Minimum Stable Random Packings of Platonic Solids

TL;DR

This study measures the stable random and densest packings of all Platonic solids, revealing that cube packings are densest among them and that packing density decreases with fewer sides, influenced by particle friction.

Contribution

It provides the first systematic experimental comparison of packing densities for all Platonic solids under different protocols, highlighting the effect of shape and friction.

Findings

Cube has the highest packing density among Platonic solids.

Packing density decreases with the number of sides of the solid.

Higher friction leads to lower packing density in tetrahedra.

Abstract

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 $\phi$.