Dense Packings of Equal Disks in an Equilateral Triangle: from 22 to 34 and Beyond.
1995; Electronic Journal of Combinatorics; Volume: 2; Linguagem: Inglês
ISSN
1097-1440
AutoresRonald Graham, Boris D. Lubachevsky,
Tópico(s)Point processes and geometric inequalities
ResumoPreviously 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 ∆(k) := k(k+1) 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
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