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arXiv · 1110.4775

Regular packings on periodic lattices

Tadeus Ras·Rolf Schilling·Martin Weigel
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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 ϕ_d(X). It is proved to be continuous with an infinite number of singular points X^{\rm min}_ν, X^{\rm max}_ν, ν=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 ϕ_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.

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Tadeus Ras, Rolf Schilling, Martin Weigel. 2011-10-21. Regular packings on periodic lattices. https://doi.org/10.1103/physrevlett.107.215503

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