Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Repeated Patterns of Dense Packings of Equal Disks in a Square. | Repeated Patterns of Dense Packings of Equal Disks in a Square R. L. Graham rlg@ B. D. Lubachevsky bdl@ AT T Bell Laboratories Murray Hill New Jersey 07974 USA Submitted January 17 1996 Accepted April 20 1996 ABSTRACT 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 .n k . Extending and improving previous work of Nurmela and Ostergard NO where previous patterns for n n k of the form k2 k2 1 k2 3 k k 1 and 4k2 k were observed we identify new patterns for n k2 2 and n k2 _k 2 . We also find denser packings than those in NO 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 .n k . which corresponds to some specific repeated pattern we identify a threshold index k0 for which the packing appears to be optimal for k k0 but for which the packing is not optimal or does not exist for k k0. 1. Introduction In a previous paper GL1 the authors observed the unexpected occurrence of repeating patterns of dense and presumably optimal packings of n equal non-overlapping disks inside an equilateral triangle see Fig. for and example . It is natural to investigate this phenomenon for other boundary shapes. In particular this was done by the authors LG1 for the case of n disks in a circle. However in contrast to the case of the equilateral triangle where the patterns appear to persist for arbitrarily large values of n for the circle the identified packing patterns cease to be optimal as the number of disks exceeds a certain threshold. In this note we describe the situation for the square. In a recent paper Nurmela and Ostergard NO present various conjectured optimal packings of n equal disks in a square for up to 50 disks. They also point out certain patterns that occur there. By using a packing procedure .
