Packing circles in a square: A review and new results

Abstract

There are many interesting optimization problems associated with the packing and covering of objects in a closed volume or bounded surface Typical examples arise in classical physics or chemistry where questions of the kind what does the densest packing of atoms or molecules look like when a crystal or macro molecule is formed with the lowest energy Also engineering and information science confront us with extremal problems associated with the packing and covering of objects One of the most prominent examples arises from the study and design of spherical codes A spherical code is a set of real vectors on the surface of the unit sphere in n dimensional Euclidian space In this case one searches for an arrangement such that the minimum separating angle between the vectors becomes as large as possible One is interested in a solution for these problems since spherical codes have important applications in the eld of information processing However it was another closely related problem the optimal packing of n equal circles in a square which has fascinated mathematicians over the last few years The circle packing problem is equivalent to the problem of scattering n points in a unit square such that the minimum distance m between any two of them becomes as large as possible The relation between the maximum radius r of the circles and the scattering distance m between the points is then given by r m