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The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary…
One emerging approach for the fabrication of complex architectures on the nanoscale is to utilize particles customized to intrinsically self-assemble into a desired structure. Inverse methods of statistical mechanics have proven…
Shape-morphing structures possess the ability to change their shapes from one state to another, and therefore, offer great potential for a broad range of applications. A typical paradigm of morphing is transforming from an initial…
The aim of this paper is to review and discuss qualitatively some results on the properties of amorphous packings of hard spheres that were recently obtained by means of the replica method. The theory gives predictions for the equation of…
We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles,…
We study growth limited by packing for irregular objects in two dimensions. We generate packings by seeding objects randomly in time and space and allowing each object to grow until it collides with another object. The objects we consider…
Microstructural materials design is one of the most important applications of inverse modeling in materials science. Generally speaking, there are two broad modeling paradigms in scientific applications: forward and inverse. While the…
We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex…
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds…
We present measurements of the stress response of packings formed from a wide range of particle shapes. Besides spheres these include convex shapes such as the Platonic solids, truncated tetrahedra, and triangular bipyramids, as well as…
Recent progress in the field of robotic manipulation has generated interest in fully automatic object packing in warehouses. This paper proposes a formulation of the packing problem that is tailored to the automated warehousing domain.…
Designing physical artifacts that serve a purpose - such as tools and other functional structures - is central to engineering as well as everyday human behavior. Though automating design has tremendous promise, general-purpose methods do…
We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and…
Colloidal self-assembly -- the spontaneous organization of colloids into ordered structures -- has been considered key to produce next-generation materials. However, the present-day staggering variety of colloidal building blocks and the…
This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container…
The three-dimensional bin packing problem (3D-BPP) plays an important role in city logistics and manufacturing environments, due to its direct relevance to operational cost. Most existing literature have investigated the conventional…
We study, via the replica method of disordered systems, the packing problem of hard-spheres with a square-well attractive potential when the space dimensionality, d, becomes infinitely large. The phase diagram of the system exhibits…