Superdense Crystal Packings of Ellipsoids

TL;DR

This paper discovers new crystal packings of ellipsoids that surpass the density of sphere packings, achieving a maximum packing fraction of approximately 0.7707, and explores their geometric properties.

Contribution

It introduces the densest known crystal packings of congruent ellipsoids, exceeding previous sphere packing densities, and identifies specific aspect ratios with maximum packing fractions.

Findings

Maximum packing density of approximately 0.7707 achieved.

Ellipsoids with aspect ratios of √3 and 1/√3 reach maximum density.

Each ellipsoid in the packing touches 14 neighbors.

Abstract

Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic lattice provides the densest packing of equal spheres with a packing fraction $\phi\approx0.7405$ \cite{Kepler_Hales}. Here we report on the densest known packings of congruent ellipsoids. The family of new packings are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids, and always surpass the densest lattice packing. A remarkable maximum density of $\phi\approx0.7707$ is achieved for both prolate and oblate ellipsoids with aspect ratios of $\sqrt{3}$ and $1/\sqrt{3}$, respectively, and each ellipsoid has 14 touching neighbors. Present results do not exclude the possibility that even denser crystal packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids.