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Kissing#d=4(Musin)

Kissing#d=4(Musin) - arXiv:math/0309430v3[math.MG 4 Oct...

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arXiv:math/0309430v3 [math.MG] 4 Oct 2006 THE KISSING NUMBER IN FOUR DIMENSIONS 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 Sch¨utte 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 n 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 moved 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 Sch¨utte and van der Waerden in 1953 [31]. A subsequent two-page sketch of a proof was given by Leech [22] in 1956. The Institute for Mathematical Study of Complex Systems, Moscow State University, Russia [email protected] 1

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thirteen spheres problem continues to be of interest, and several new proofs have been published in the last few years [20, 24, 6, 1, 26]. Note that k (4) 24. Indeed, the unit sphere in R 4 centered at (0 , 0 , 0 , 0) has 24 unit spheres around it, centered at the points ( ± 2 , ± 2 , 0 , 0), with any choice of signs and any ordering of the coordinates. The convex hull of these 24 points yields a famous 4-dimensional regular polytope - the “24-cell”. Its facets are 24 regular octahedra. Coxeter proposed upper bounds on k ( n ) in 1963 [10]; for n = 4 , 5 , 6 , 7 , and 8 these bounds were 26, 48, 85, 146, and 244, respectively. Coxeter’s bounds are based on the conjecture that equal size spherical caps on a sphere can be packed no denser than packing where the Delaunay triangulation with vertices at the centers of caps consists of regular simplices. This conjecture has been proved by B¨or¨ oczky in 1978 [5].
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Kissing#d=4(Musin) - arXiv:math/0309430v3[math.MG 4 Oct...

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