The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden.In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4) = 24 is proved. The proof is based on a modification of Delsarte's method.
A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g(n) of spherical two-distance sets does not exceed n(n + 3)/2. This upper bound is known to be tight for n = 2, 6, 22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n) = n(n + 1)/2 for g(n).In this paper using the so-called polynomial method it is proved that for nonnegative a + b the largest cardinality of S is not greater than L(n). For the case a + b < 0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n) = L(n) for 6 < n < 22, 23 < n < 40, and g(23) = 276 or 277.
The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schütte and van der Waerden only in 1953.A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.
The Tammes problem is to find the arrangement of N points on a unit sphere which maximizes the minimum distance between any two points. This problem is presently solved for several values of N , namely for N = 3, 4, 6, 12 by L. Fejes Tóth (1943); for N = 5, 7, 8, 9 by Schütte and van der Waerden (1951); for N = 10, 11 by Danzer (1963) and for N = 24 by Robinson (1961). Recently, we solved the Tammes problem for N = 13.The optimal configuration of 14 points was conjectured more than 60 years ago. In the paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof relies on an enumeration of the irreducible contact graphs. *
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