Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing problem can be cast as a nonconvex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm ...

We consider packings of the plane using discs of radius 1 and r=0.545151... . The value of r admits compact packings in which each hole in the packing is formed by three discs which are tangent to each other. We prove that the largest density possible is that of the compact packing shown in figure 1.

We provide a counterexample to a conjecture by B. Connelly about density of circle packings

A brief report on recent work on the sphere-packing problem.

We have studied the packing of congruent disks on a spherical cap, for caps of different size and number of disks, $N$. This problem has been considered before only in the limit cases of circle packing inside a circle and on a sphere (Tammes problem), whereas all intermediate cases are unexplored. Finding the preferred packing configurations for a domain with both curvature and border could be useful in the description of physical and biological systems (for example, colloida...

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 quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle pac...

In this note we prove that, if $S_n$ is the greatest area of a rectangle which can be covered with $n$ unit disks, then $2\leq S_n/n<3 \sqrt{3}/2$, and these are the best constants; moreover, for $\Delta(n):=(3\sqrt{3}/2)n-S_n$, we have $0.727384<\liminf\Delta(n)/\sqrt{n}<2.121321$ and $0.727384<\limsup\Delta(n)/\sqrt{n}<4.165064$.

A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral tr...