Dense Packings of Polyhedra: Platonic and Archimedean Solids

TL;DR

This paper introduces the Adaptive Shrinking Cell scheme to find dense packings of non-tiling polyhedra, revealing new densest packings for several Platonic solids and proposing conjectures on optimal packing structures.

Contribution

It formulates a novel optimization approach for dense packings of polyhedra and provides the densest known packings for various Platonic solids, along with theoretical bounds and conjectures.

Findings

Densest packings of tetrahedra, icosahedra, dodecahedra, and octahedra identified.

Densest tetrahedral packing has no long-range order.

Conjecture: densest packings of centrally symmetric solids are their densest lattice packings.

Abstract

We formulate the problem of generating dense packings of nonoverlapping, non-tiling 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 the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823, 0.836, 0.904, and 0.947, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles, and apply it to Platonic solids, Archimedean solids, superballs and ellipsoids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing.