Lattice packings with gap defects are not completely saturated

TL;DR

The paper proves that certain lattice packings with linear gap defects, such as honeycomb circle packings and fcc sphere packings with planar gaps, cannot be completely saturated, addressing open problems in packing theory.

Contribution

It demonstrates that specific lattice packings with linear gap defects are inherently not completely saturated, extending understanding of packing limitations with defects.

Findings

Honeycomb circle packings with linear gaps are not completely saturated.

fcc sphere packings with planar gaps are not completely saturated.

Addresses open problems in the theory of saturated packings with defects.

Abstract

We show that a honeycomb circle packing in $\R^2$ with a linear gap defect cannot be completely saturated, no matter how narrow the gap is. The result is motivated by an open problem of G. Fejes T\'oth, G. Kuperberg, and W. Kuperberg, which asks whether of a honeycomb circle packing with a linear shift defect is completely saturated. We also show that an fcc sphere packing in $\R^3$ with a planar gap defect is also not completely saturated.