Lattice packings with gap defects are not completely saturated

Abstract

We show that a honeycomb circle packing in ${\mathbb{R}}^2$ 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 Tóth, 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 ${\mathbb{R}}^3$ with a planar gap defect is also not completely saturated.

Cited by in corpus (2)

• Geometric Frustration and Interparticle Gap Size Distributions in Ordered Hexagonal Polydisperse Disk Packs
• On densest packings of equal balls of $\rb^{n}$ and Marcinkiewicz spaces