Circle packings

Hexagonal packing of equal circles has been known since early antiquity (e.g. the fourth picture on page 43). It fills a fraction π/√12 ≃ 0.91 of area—which was proved maximal for periodic packings by Carl Friedrich Gauss in 1831 and for any packing by Axel Thue in 1910 and László Fejes Tóth in 1940. Much has been done to study densest packings of limited numbers of circles into various shapes, as well as onto surfaces of spheres (as in golf balls, pollen grains or radiolarians). Typically it has been found that with enough circles, patches of hexagonal packing always tend to form. (See page 987.)

For circles of unequal sizes rather little has been done. A procedure analogous to the one on page 350 was introduced by Charles Bennett in 1971 for 3D spheres (relevant for binary alloys). The picture below shows the network of contacts between circles in the cases from page 350. Note that with the procedure used, each new circle added must immediately touch two existing ones, though subsequently it may get touched by varying numbers of other circles.

The distribution of numbers of circles that touch a given circle changes with the ratio of circle sizes, as in the picture below. The total filling fraction seems to vary fairly smoothly with this ratio, though I would not be surprised if some small-scale jumps were present.

Note that even a single circle of different size in the center can have a large-scale effect on the results of the procedure, as illustrated in the pictures below.

Finding densest packings of n circles is in general like solving quadratic programming problems with about n^{2} constraints. But at least for many size ratios I suspect that the final result will simply involve each kind of circle forming a separated hexagonally-packed region. This will not happen, however, for size ratios ≤ 2/√3 - 1 ≃ 0.15, since then the small circles can fit into the interstices of an ordinary hexagonal pattern, yielding a filling fraction 1/18(17 √13 - 24) π ≃ 0.95. The picture below shows what happens if one repeatedly inserts circles to form a so-called Apollonian packing derived from the problem studied by Apollonius of finding a circle that touches three others. At step t, 3^{t-1} circles are added for each original circle, and the network of tangencies among circles is exactly example (a) from page 509. Most of the circles added at a given step are not the same size, however, making the overall geometry not straightforwardly nested. (The total numbers of different sizes of circles for the first few steps are {2, 3, 5, 10, 24, 63, 178, 521}. At step 3, for example, the new circles have radii (25 - 12 √3)/193 and (19 - 6 √3)/253. In general, the radius of a circle inscribed between three other touching circles that have radii p, q, r is p q r/(p q + p r + q r + 2 Sqrt[p q r (p + q + r)]).) In the limit of an infinite number of steps the filling fraction tends to 1, while the region left unfilled has a fractal dimension of about 1.3057.

To achieve filling fraction 1 requires arbitrarily small circles, but there are many different arrangements of circles that will work, some not even close to nested. When actual granular materials are formed by crushing, there is probably some tendency to generate smaller pieces by following essentially substitution system rules, and the result may be a nested distribution of sizes that allows an Apollonian-like packing.

Apollonian packings turn out to correspond to limit sets invariant under groups of rational transformations in the complex plane. Note that as on page 1007 packings can be constructed in which the sizes of circles vary smoothly with position according to a harmonic function.