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Lattice packings with gap defects are not completely saturated. (English) Zbl 1054.52010

A packing of unit spheres in \({\mathbb R}^n\) is \(k\)-saturated if it is not possible to replace \(k-1\) spheres by \(k\) spheres and still have a packing, and completely saturated if it is \(k\)-saturated for every \(k\).
Considering the honeycomb packing in \({\mathbb R}^2\) and the fcc packing in \({\mathbb R}^3\) with linear (resp. planar) gap, the authors prove that rearrangement of finitely many circles (spheres) is possible to ensure enough space for a new circle (sphere). This means that such packings are not completely saturated. Some open questions in this area are listed.

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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