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Báo cáo toán học: "Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond. | Dense Packings of Equal Disks in an Equilateral Triangle From 22 to 34 and Beyond R. L. Graham B. D. Lubachevsky AT T Bell Laboratories Murray Hill New Jersey 07974 Submitted August 11 1994 Accepted December 7 1994 ABSTRACT Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up to 34 disks. For each n in the range 22 n 34 we present what we believe to be the densest possible packing of n equal disks in an equilateral triangle. For these n we also list the second often the third and sometimes the fourth best packings among those that we found. In each case the structure of the packing implies that the minimum distance d n between disk centers is the root of polynomial Pn with integer coefficients. In most cases we do not explicitly compute Pn but in all cases we do compute and report d n to 15 significant decimal digits. Disk packings in equilateral triangles differ from those in squares or circles in that for triangles there are an infinite number of values of n for which the exact value of d n is known namely when n is of the form A k ki k Vi. It has also been conjectured that d n 1 d n in this case. Based on our computations we present conjectured optimal packings for seven other infinite classes of n namely n A 2k 1 A 2k 1 1 A k 2 2 A 2k 3 3 A 3k 1 2 4A k and 2A k 1 2A k 1 . We also report the best packings we found for other values of n in these forms which are larger than 34 namely n 37 40 42 43 46 49 56 57 60 63 67 71 79 84 92 93 106 112 121 and 254 and also for n 58 95 108 175 255 256 258 and 260. We say that an infinite class of packings of n disks n n 1 n 2 .n k . is tight if 1 d n k 1 1 d n k is bounded away from zero as k goes to infinity. We conjecture that some of our infinite classes are tight others are not tight and that there are infinitely many tight classes. THE ELECTRONIC JOURNAL OF COMBINATORICS 2 1995 A1 1 Introduction 2