Sphere Packings pptx

Sphere Packings Chuanming Zong Springer [...]... Multiple Sphere Packings 10.1 Introduction 10.2 A Basic Theorem of Asymptotic Type 10.3 A Theorem of Few and Kanagasahapathy 10.4 Remarks on Multiple Circle Packings 153 153 154 157 162 11 Holes in Sphere Packings 11.1 Spherical Holes in Sphere. .. of Kepler’s conjecture Clearly, the density of any sphere packing may be improved by adding spheres as long as there is sufficient room to do so When there is no longer room to add additional spheres we say that the sphere packing is 14 1 The Gregory-Newton Problem, Kepler’s Conjecture saturated Without loss of generality, we assume that the sphere packings considered in the following subsections are... 153 153 154 157 162 11 Holes in Sphere Packings 11.1 Spherical Holes in Sphere Packings 11.2 Spherical Holes in Lattice Sphere Packings 11.3 Cylindrical Holes in Lattice Sphere Packings 165 165 176 178 12 Problems of Blocking Light Rays 12.1 Introduction ... sometimes referred to as the thirteen spheres problem In S3 + Λ3 , locally speaking, the kissing configuration is stable In other words, none of the twelve spheres that touch S3 can be moved around (see Figure 1.3) However, if twelve unit spheres are placed at positions corresponding to the vertices of a regular icosahedron concentric with the central one, the twelve outer spheres do not touch each other... Packing Densities and the Kissing Numbers of Spheres I 91 6.1 Blichfeldt’s Upper Bound for the Packing Densities of Spheres 91 6.2 Rankin’s Upper Bound for the Kissing Numbers of Spheres 95 6.3 An Upper Bound for the Packing Densities of Superspheres ... Barlow [1] found infinitely many nonlattice packings of spheres with the same density They are the laminations of hexagonal layers of spheres Removing the lattice restriction, the first proof of π δ(S2 ) = √ 12 (1.27) was achieved by Thue [1] and [2] Roughly speaking, Thue’s idea was to compute the area left uncovered by the circular disks in certain finite packings His method was developed further by... Kissing Numbers of Spheres, and Watson’s Theorem 23 25 31 33 36 41 xii Contents 2.7 Three Mathematical Geniuses: Zolotarev, Minkowski, and Voronoi 3 4 Lower Bounds for the Packing Densities of Spheres 3.1 The Minkowski-Hlawka Theorem 3.2 Siegel’s Mean Value Formula 3.3 Sphere Coverings and... of Spheres 4.1 The Blocking Numbers of S3 and S4 4.2 The Shannon-Wyner Lower Bound for Both b(Sn ) and k(Sn ) 4.3 A Theorem of Swinnerton-Dyer 4.4 A Lower Bound for the Translative Kissing Numbers of Superspheres 42 47 47 51 55 62 65 65 71 72 74 5 Sphere. .. face-centered cubic lattice It is easy to see that S3 + Λ3 is a packing of S3 , in which every sphere touches 12 others This observation implies k ∗ (S3 ) ≥ 12 (1.17) In 1694, during a famous conversation, D Gregory and I Newton discussed the following problem The Gregory-Newton Problem Can a sphere touch 13 spheres of the same size? Newton thought “no, the maximum number is 12,” while Gregory believed... Introduction Packings of Circular Disks The Gregory-Newton Problem Kepler’s Conjecture Some General Remarks 1 7 10 13 18 Positive Definite Quadratic Forms and Lattice Sphere Packings 23 2.1 Introduction . Multiple Circle Packings 162 11. Holes in Sphere Packings 165 11.1. Spherical Holes in Sphere Packings . . . . . . . . . . . . . . . . 165 11.2. Spherical Holes in Lattice Sphere Packings . . deals not only with the classical sphere packing problems, but also the contemporary ones such as blocking light rays, the holes in sphere packings, and finite sphere packings. Not only are the main results. left blank Preface Sphere packings is one of the most fascinating and challenging subjects in mathematics. Almost four centuries ago, Kepler studied the densities of sphere packings and made his