Let s n be the side of the smallest square into which it is possible to pack n congruent squares. In this paper we link s n to the supremum of the maximal inflation ω(C) of admissible configurations C. The computation and the properties of ω(C) and related functions give rise to an algorithm similar to the billiard approach used to pack congruent disks or spheres in a bounded domain. We improve the best known packings of n equal squares for n = 11, 29 and 37, and give an alternative optimal packing of 18 squares.
10 figures 10th IMACS International Symposium on Iterative Methods in Scientific ComputingInternational audienceThis paper addresses the classical and discrete Euler-Lagrange equations for systems of $n$ particles interacting quadratically in $\mathbb{R}^d$. By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. The roots of classical and discrete QEP being given, we state some conditional convergence results. Next, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence
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