2011/10/21 by Tadeus Ras, R. Schilling, Rolf Schilling +1
Computer Science · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Atomic packing factor #Combinatorics #Computational Geometry and Mesh Generation #Computer science #Condensed matter physics #Context (archaeology) #Ellipse #Ellipsoid #Ising model #Lattice (music) #Mathematics #Maxima #Optimization and Packing Problems #Physics #Quasicrystal Structures and Properties #Space (punctuation) #Square lattice #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1103/physrevlett.107.215503
published as Phys. Rev. Lett. 107, 215503 (2011) · 5 pages, 4 figures, accepted for publication in Physical Review Letters
arxiv created 2011/10/21 · openalex publication_date 2011/11/18 · arxiv updated 2011/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \ensuremathφd(X). It is proved to be continuous with an infinite number of singular points X_\ensuremathνmin,X_\ensuremathνmax, \ensuremathν=0,\ifmmode±\else\textpm\fi1,\ifmmode±\else\textpm\fi2,…. In two dimensions, all maxima have the same height, whereas there is a unique global maximum for the case of ellipsoids. The form of \ensuremathφd(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.