×

Dense packings of equal disks in an equilateral triangle: From 22 to 34 and beyond. (English) Zbl 0817.52020

Electron. J. Comb. 2, Article A1, 39 p. (1995); printed version J. Comb. 2, 1-39 (1995).
Summary: 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 \(22 \leq n \leq 34\) 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 packing 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 a polynomial \(P_ n\) with integer coefficients. In most cases we do not explicitly compute \(P_ n\) 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 \(\Delta (k) : = {k(k + 1) \over 2}\). 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 \[ \begin{aligned} n = & \Delta (2k) + 1,\;\Delta (2k + 1) + 1,\;\Delta (k + 2) - 2,\;\Delta (2k + 3) - 3,\\ & \Delta (3k + 1) + 2,\;4 \Delta (k), \text{ and } 2 \Delta (k + 1) + 2 \Delta (k) - 1. \end{aligned} \] 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, \(n = n(1)\), \(n(2), \dots, n(k), \dots\), 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.

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
05B40 Combinatorial aspects of packing and covering
PDFBibTeX XMLCite
Full Text: arXiv EuDML EMIS