Kissing numbers, sphere packings, and some unexpected proofs

The ikissing number problemi asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are classical, but the answers in dimensions eight and twenty-four were a big surprise in 1979, based on an extremely elegant method initiated by Philippe Delsarte in the early seventies, which concerns inequalities for the distance distributions of kissing congurations. Delsarte’s approach led to especially striking results in cases where there are exceptionally symmetric, dense and unique congurations of spheres: In dimensions eight and twenty-four these are given by the shortest vectors in two remarkable lattices, known as the E8 and the Leech lattice. However, despite the fact that in dimension four there is a special conguration which is conjectured to be optimal and uniqueothe shortest vectors in the D4 lattice, which are also the vertices of a regular 24-celloit was proved that the bounds given by Delsarte’s method aren’t good enough to solve the problem in dimension four. This may explain the astonishment even to experts when in the fall of 2003, Oleg Musin announced a solution of the problem, based on a clever modication of Delsarte’s method [21, 22].

[1]  Thomas M. Thompson.  From error-correcting codes through sphere packings to simple groups , 1983 .

[2]  R. A. Cuninghame-Green,et al.  Packing and Covering in Combinatorics , 1980 .

[3]  George G. Szpiro.  Kepler's Conjecture and Hales's Proof , 2003 .

[4]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[5]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[6]  Олег Рустумович Мусин,et al.  Проблема двадцати пяти сфер@@@The problem of the twenty-five spheres , 2003 .

[7]  V. Arestov,et al.  Estimates of the maximal value of angular code distance for 24 and 25 points on the unit sphere in ℝ4 , 2000 .

[8]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[9]  田上 真.  Oleg Musinの論文「The kissing number in four dimensions 」の紹介 (代数的組合せ論) , 2004 .

[10]  William H. Press,et al.  Numerical recipes in C , 2002 .

[11]  N. J. A. Sloane,et al.  New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions , 1979, J. Comb. Theory, Ser. A.

[12]  Ralph Duncan James,et al.  Proceedings of the International Congress of Mathematicians , 1975 .

[13]  P. Delsarte.  Bounds for unrestricted codes, by linear programming , 1972 .

[14]  T. Hales.  Cannonballs and Honeycombs , 2000 .

[15]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[16]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[17]  George E. Andrews,et al.  Special Functions: Partitions , 1999 .

[18]  Henry Cohn,et al.  New upper bounds on sphere packings I , 2001, math/0110009.

[19]  O. Musin.  The kissing number in four dimensions , 2003, math/0309430.

[20]  Martin Aigner,et al.  The problem of the thirteen spheres , 1998 .

[21]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[22]  Thomas C. Hales.  A computer verification of the Kepler conjecture , 2003 .

[23]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[24]  B. L. Waerden,et al.  Das Problem der dreizehn Kugeln , 1952 .

[25]  Volker Schönefeld.  Spherical Harmonics , 2019, An Introduction to Radio Astronomy.

[26]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[27]  J. Seidel,et al.  Spherical codes and designs , 1977 .

[28]  Henry Cohn,et al.  The densest lattice in twenty-four dimensions , 2004, math/0408174.

[29]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .