tailieunhanh - Báo cáo toán học: "Dense Packings of Equal Spheres in a Cube"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Dense Packings of Equal Spheres in a Cube. | Dense Packings of Equal Spheres in a Cube Th. Gensane Laboratoire de Mathematiques Pures et Appliquees Joseph Liouville 50 rue F. Buisson bP 699 62228 Calais Cedex FRANCE gensane@ Submitted Jun 6 2003 Accepted May 20 2004 Published May 27 2004 MR Subject Classification 52C17 05B40. Abstract We describe an adaptation of the billiard algorithm for finding dense packings of equal spheres inside a domain of the euclidean space. In order to improve the convergence of this stochastic algorithm we introduce systematic perturbations in it. We apply this perturbed billiard algorithm in the case of n spheres in a cube and display all the optimal and best known packings up to n 32. We improve the previous record packings for all 11 n 26 except n 13 14 18. We prove the existence of the displayed packings for n 11 12 15 17 20 21 22 26 32 by constructing them explicitly. For example the graph of the conjectured optimal packing of twenty-two spheres is composed of five octahedrons and four isolated points. We also conjecture that the minimum distance dn between spheres centers of the optimal packings is constant in the range 29 n 32. 1 Introduction We first consider the maximum separation problem in a domain K of the euclidean space Ed how to spread n points p1 pn inside K so that the minimum distance is as large as possible. This formulation is often used by the authors dealing with the disk packing problem which consists in determining the maximum diameter of n equal non-overlapping disks contained in K. A discussion about the equivalence between the two problems can be found in M . For other presentations of the subject and related questions see CFG SK T Z . It appears in the mathematical literature that there is no domain where the solution of the problem is obvious. Moreover it is significant that the regular hexagonal packings of n fc fc2 1 disks in the equilateral triangle compose the only infinite family of optimal packings known and proved O . We .