On regular and random two-dimensional packing of crosses

TL;DR

This study investigates the packing behavior of 2D crosses, comparing random packings with regular structures, and analyzes local correlations and orientational order in jammed configurations.

Contribution

It provides experimental data on the packing fractions and local correlations of randomly packed 2D crosses with various aspect ratios, a problem not previously well-understood.

Findings

Random packing fractions vary with aspect ratio.

Local orientational order extends over 2-3 cross lengths.

Spatial correlations of neighbors also extend over 2-3 crosses.

Abstract

Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packings are known. The packing of 2D crosses does not yet belong to the category of solved problems. We demonstrate in experiments with crosses of different aspect ratios (arm width to length) which packing fractions are actually achieved by random packing, and we compare them to densest regular packing structures. We determine local correlations of the orientations and positions after ensembles of randomly placed crosses were compacted in the plane until they jam. Short-range orientational order is found over 2 to 3 cross lengths. Similarly, correlations in the spatial distributions of neighbors extend over 2 to 3 crosses. Apparently, there is no simple relation between the geometries of the crosses and peaks in the spatial correlation functions. Some features of the orientational correlations, however, are intuitively evident.