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Repeated patterns of dense packings of equal disks in a square. (English) Zbl 0851.05038

Electron. J. Comb. 3, No. 1, Research paper R16, 17 p. (1996); printed version J. Comb. 3, No. 1, 211-227 (1996).
Summary: We examine sequences of dense packings of \(n\) congruent non-overlapping disks inside a square which follow specific patterns as \(n\) increases along certain values, \(n= n(1), n(2),\dots, n(k),\dots\). Extending and improving previous work of Nurmela and Östergård where previous patterns for \(n= n(k)\) of the form \(k^2\), \(k^2- 1\), \(k^2- 3\), \(k(k+ 1)\), and \(4k^2+ k\) were observed, we identify new patterns for \(n= k^2- 2\) and \(n= k^2+ \lfloor k/2\rfloor\). We also find denser packings than those in Nurmela and Östergård for \(n= 21, 28, 34, 40, 43, 44, 45\), and 47. In addition, we produce what we conjecture to be optimal packings for \(n= 51, 52, 54, 55, 56, 60\), and 61.
Finally, for each identified sequence \(n(1), n(2),\dots, n(k),\dots\) which corresponds to some specific repeated pattern, we identify a threshold index \(k_0\), for which the packing appears to be optimal for \(k\leq k_0\), but for which the packing is not optimal (or does not exist) for \(k> k_0\).

MSC:

05B40 Combinatorial aspects of packing and covering
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)