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 longstanding 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 NewtonGregory 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 twopage 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 4dimensional regular polytope  the “24cell”. 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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