Kissing number problem

In geometry, the - te kiss number is (even contact number ) the maximum number of -dimensional unit spheres, ie spheres of radius 1 that can touch another such unit sphere in Euclidean space simultaneously without overlaps occur. Additionally, the condition may be assumed that the centers of the balls must be in a lattice (grid kiss numbers). As Kiss number problem the lack of a general formula for calculating the kiss numbers is known.

Kiss figures in various dimensions

In one dimension, the unit sphere is a track whose end points are at a distance 1 from the origin. Here, at both endpoints, respectively, an additional route to be added, so that the kissing number for a dimension is obviously 2.

In the second dimension, the unit sphere is a circle with radius 1 Clearly corresponds to the problem of determining the number kiss in this dimension of the task, as many coins to arrange it so that they touch all an equal central coin. It is easy to see ( and prove ) that the kissing number for the second dimension 6.

In the third dimension, the calculation is not so simple; see the diagram on the right. It is easy to arrange twelve balls so that they touch the central sphere ( for example, so that their centers form the corners of a cuboctahedron ). But this case remains a lot of space, and it is not obvious that this is not enough space to add a thirteenth ball. In fact, so much space is available, that any two beads can change their places from the twelve outer, without losing contact with the central ball. This issue was the subject of a famous dispute between the mathematicians Isaac Newton and David Gregory, both of which led at a discussion to Kepler 's conjecture in 1692. Newton claimed the maximum is twelve, Gregory said it was thirteen. In the 19th century, the first publications that claimed to contain evidence of Newton's assertion appeared. By today's standards, formal proofs, however, were performed in 1953 by Kurt Bartel Leendert van der Schütte and Waerden and 1956 by John Leech.

Only at the beginning of the 21st century has been proved that the kissing number for the fourth dimension is 24.

Further, the kiss figures for the dimensions (240) and ( 196 560 ) are known; in 24- dimensional space, the balls are placed on the points of the Leech lattice, so that no space is left. The following table lists the known limits for the kissing number up to size 24 again.

Estimates show that the growth of Kiss figures is exponential; see chart next to the table. The base of the exponential growth is unknown. The exact Kiss figures for the dimensions 8 and 24 were determined independently in 1979 by Andrew M. Odlyzko and Neil JA Sloane and Vladimir Levenshtein.

Rather little is known about the kissing numbers in even higher dimensions; lower bounds are known about the dimensions ( 276 032 ) ( 438 872 ) ( 991 792 ) ( 2,948,552 ) ( 331 737 984 ) and ( 1368532064 ).

Grid Kiss figures in various dimensions

The exact grating kiss numbers are known for the dimensions of 1 to 9 and for the dimension 24. The following table gives the lattice Kiss figures and the known lower bounds up to size 24 again:

The lattice packings for dimensions 12 and 24 have their own name: The Coxeter - Todd lattice ( according to Harold Scott MacDonald Coxeter and John Arthur Todd ) for the dimension 12 and the Leech lattice ( by John Leech ) for the dimension 24

The general shape of the lower limit for dimensional grid metrics is given by

The Riemann zeta function. This limit is specified by the set of Minkowski Hlawka (after Hermann Minkowski and Edmund Hlawka ).