TAILIEUCHUNG - Đề tài " The kissing number in four dimensions "

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 Sch¨tte and van der u 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. | Annals of Mathematics The kissing number in four dimensions By Oleg R. Musin Annals of Mathematics 168 2008 1 32 The kissing number in four dimensions By Oleg R. Musin Abstract 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 Schutte 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. 1. Introduction The kissing number k n is the highest number of equal nonoverlapping spheres in R that can touch another sphere of the same size. In three dimensions the kissing number problem is asking how many white billiard balls can kiss touch a black ball. The most symmetrical configuration 12 billiard balls around another is if the 12 balls are placed at positions corresponding to the vertices of a regular icosahedron concentric with the central ball. However these 12 outer balls do not kiss each other and may all move freely. So perhaps if you moved all of them to one side a 13th ball would possibly fit in This problem was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. It is commonly said that Newton believed the answer was 12 balls while Gregory thought that 13 might be possible. However Casselman 8 found some puzzling features in this story. The Newton-Gregory problem is often called the thirteen spheres problem. Hoppe 18 thought he had solved the problem in 1874. However there was a mistake an analysis of this mistake was published by Hales 17 in 1994. Finally this problem was solved by Schutte and van der Waerden in 1953 31 . 2 OLEG R. MUSIN A subsequent two-page sketch

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