Regular packings on periodic lattices

TL;DR

This paper studies the densest packings of identical objects on regular lattices in various dimensions, analyzing how packing density varies with aspect ratio and identifying singular points and maximum packing configurations.

Contribution

It provides a detailed analysis of maximum packing fractions for rectangles and ellipsoids on square and cubic lattices, including proofs of continuity and identification of singular points.

Findings

Maximum packing fraction is continuous with infinite singular points.

All maxima in 2D have the same height; 3D ellipsoids have a unique global maximum.

The form of packing fraction relates to geometrical frustration and number theory.

Abstract

We investigate the problem of packing identical hard objects on regular lattices in d dimensions. Restricting configuration space to parallel alignment of the objects, we study the densest packing at a given aspect ratio X. For rectangles and ellipses on the square lattice as well as for biaxial ellipsoids on a simple cubic lattice, we calculate the maximum packing fraction \phi_d(X). It is proved to be continuous with an infinite number of singular points X^{\rm min}_\nu, X^{\rm max}_\nu, \nu=0, \pm 1, \pm 2,... In two dimensions, all maxima have the same height, whereas there is a unique global maximum for the case of ellipsoids. The form of \phi_d(X) is discussed in the context of geometrical frustration effects, transitions in the contact numbers and number theoretical properties. Implications and generalizations for more general packing problems are outlined.