Packing Circles in a Square A Review and New Results

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 m It is very surprising that such a problem which at rst looks rather simple has brought to us a series of nice papers with a continuous improvement of the results In this talk we give a short review on the history and the well known facts on the packing problem of n equal circles in a square referencing recently obtained results We present new results including the proof of optimality for up to n circles And we calculate the exact expression for the optimal scattering distance obtained within a computer algebraic approach More precisely we give the minimal polynomial corresponding to the optimal solution In the following we discuss and summarize our ndings n The situation for packing of up to circles in a square was already solved in Here the cases n and are solved easily For n it was R L Graham mentioned in who found the optimal solution The proofs for n and n were done by J Schaer the one for n by J Schaer and A Meir Interesting about the optimal packing of circles is the fact that one circle is free i e its center can be moved within a bounded region n This problem has a long history which began in when M Goldberg proposed a symmetric arrangement consisting of rows of circles The circles in this packing have a radius r In J Schaer increased the radius to r even though his packing contains two free circles in opposite corners of the square Sixteen years later R Milano proved that his packing with r is the best symmetric one But last year G Valette found a better chaotic solution which means that there is no symmetry in the arrangement of the circles in the square The radius now was r Also his solution did not survive very long In the Abstract Index of the Zbl Math we nd the following note This packing has been improved in by B Gr unbaum the radius of the circles is now approximately r We recently found a packing with a radius r and showed that it is the optimum one When this work had been nished we learned that this solution was already