Some Densest Two-Size Disc Packings in the Plane

Abstract

The main purpose of this paper is to prove some long-standing conjectures concerning the packing density of some compact arrangements of discs of two different radii in the Euclidean plane. To reach this goal a new method, called cell balancing, is presented.

Compact Packings of the Plane with Two Sizes of Discs

“…The sequence for small discs is unique and there are six sequences for larges discs: rr1r 1111(8), rr11r 111(8), rrr1r 1r 1r 1(10), rr1rr1r 1r 1(10), rr1r 1rr1r 1(10) or 111111 (6).…”

“…For six of the above values of r (c 1 , c 3 , c 4 , c 6 , c 7 , c 8 ), Heppes has proved that the densest packing is a compact packing [5], [6]. We expect that for the other three values of r the densest packing is also a compact packing.…”

The Isostatic Conjecture

“…In [26], Kennedy shows a list of nine classes of all the triangulated packings of disks in the plane with just two disk sizes. Seven of those nine packing have been shown, by Heppes and Kennedy [24,25] to be the most dense using just those two sizes, which is a bit stronger than the statement of Conjecture 12.1. This is support for Conjecture 12.1.…”

“…Notice in the case when k = 2, and r 2 /r 1 = √ 2 − 1, and n 1 = n 2 the statement of Conjecture 12.1 is weaker that Heppes's Theorem [24], since it assumes n 1 = n 2 . On the other hand, as far as we know, for other proportions of sizes of disks, Conjecture 12.1 is not known.…”

“…. , in [24] Alidár Heppes proved that the maximum density for such a packing is π(2 − √ 2)/2 = 0.92015 . .…”