Patterns and Structures in Disk Packings

Using a computational procedure that imitates tightening of an assembly of billiard balls, we have generated a number of packings of n equal and non-equal disks in regions of various shapes. Our experiments are of three major types. In the first type, the values of n are in thousands, the initial disk configuration is random and a priori one expects the generated packings to be random. In fact, the packings turn out to display non-random geometric patterns and regular features, including polycrystalline textures with "rattlers" typically trapped along the grain boundaries. An experiment of the second type begins with a known or conjectured optimal disk packing configuration, which is then "frustrated" by a small perturbation such as variation of the boundary shape or a relative increase of the size of a selected disk with respect to the sizes of the other disks. We present such frustrated packings for both large n (~ 10, 000) and small n (~ 50 to 200). Motivated by applications in material science and physics, the first and second type of experiments are performed for boundary shapes rarely discussed in the literature on dense packings: torus, a strip cut from a cylinder, a regular hexagon with periodic boundaries. Experiments of the third type involve the shapes popular among mathematicians: circles, squares, and equilateral triangles the boundaries of which are hard reflecting walls. The values of n in these experiments vary from several tens to few hundreds. Here the obtained configurations could be considered as candidates for the densest packings, rather than random ones. Some of these conjecturally optimal packings look regular and the regularity often extends across different values of n. Specifically, as n takes on an increasing sequence of values, n = n(1), n(2), ...n(k), ..., the packings follow a well-defined pattern. This phenomenon is especially striking for packings in equilateral triangles, where (as far as we can tell from our finite computational experiments), not only are there an infinite number of different patterns, each with its own different sequence n(1), n(2), ...n(k), ..., but many of these sequences seem to continue indefinitely. For other shapes, notably squares and circles, the patterns either cease to be optimal or even cease to exist (as packings of non-overlapping disks) above some threshold value n(k0) (depending on the pattern). In these cases, we try to identify the values of n(k0).     

Geometric properties of random disk packings

Random packings ofN2000 rigid disks in the plane, subject to periodic boundary conditions on a square primitive cell, have been generated by a concurrent construction which treats all disks on an equal footing, as opposed to previously investigated sequential constructions. The particles start with random positions and velocities, and as they move about they grow uniformly in size, from points to jammed disks. The collection of packings displays several striking geometric features. These include (for largeN) typically polycrystalline textures with irregular grain boundaries and linear shear fractures. The packings occasionally contain monovacancies and trapped but unjammed rattler disks. The latter appear to be confined to the grain boundaries. The linear shear fractures preserve bond orientational order, but disrupt translational order, within the crystalline grains. A new efficient event-driven simulation algorithm is employed to generate the histories of colliding and jamming disks. On a computer which can process one million floating-point instructions per second the algorithm processes more than one million pairwise collisions per hour.     

Patterns of broken symmetry in the impurity-perturbed rigid-disk crystal

As a new example of spontaneous pattern formation in many-body systems, we examine the collective means by which a close-packed disk crystal reacts to the presence of a single oversized impurity disk. Computer simulation has been used for this purpose; it creates the jammed impurity-containing packings by a kinetic particle-growth algorithm. Hexagonal primitive cells with periodic boundary conditions were employed, and the natural number 3n ² of disks (including the impurity) ranged upt to 10,800. For impurity diameter 1.2 times that of the other disks, the patterns of observed crystal perturbation displayed several remarkable features. Particle displacements relative to the unperturbed triangular crystal possess local irregularity but long-range coherence. The symmetry of the coherent patterns preserved that of the hexagonal cell for rapid growth, but was lower for slower growth. The final jammed packings contain rattler disks of the sort known to apper in random disk packings. Finally, the area increase induced by the presence of a fixed-size impurity appears to grow without bound as the system size (i.e., 3n ²) itself increases.