In the spring of 1694, Isaac Newton had a discussion with the Scottish mathematician David Gregory. The conversation took place on the campus of Cambridge University. What exactly was said is unclear, but Newton’s later published correspondence indicates that it was about how many spheres of equal size can touch a central sphere of that same size. Mathematicians later started calling that number the ‘kissing number’ – a reference to billiard balls ‘kissing’ each other when they come into contact after a bump.
Newton and Gregory would have known that the kiss number is at least twelve. Place the centers of twelve identical spheres on the twelve vertices of an icosahedron (regular twenty-hedron), and one central sphere inside that icosahedron. The twelve outer spheres then kiss that central sphere, but not each other. In fact, they have so much play that they can change places without breaking contact with the central sphere. So perhaps a thirteenth sphere would fit if you place those twelve spheres against each other as much as possible?
We will never know what Newton and Gregory thought about this, although it is speculated that Newton was convinced of the impossibility of thirteen tangent spheres, while Gregory thought it might be possible. If that reconstruction of their positions is correct, Newton was right: the kiss number in three dimensions is twelve, not thirteen. The first generally accepted evidence was not delivered until 1952by the German Kurt Schütte and the Dutchman Bartel van der Waerden.
Spheres in higher dimensions
Ordinary spheres are three-dimensional, but in the nineteenth century mathematics began to move away from concrete images and a trend towards abstraction emerged. Spheres in higher dimensions can no longer be visualized, but they can be described with coordinates. The higher the number of dimensions, the stranger their behavior. For example, the volume of an ‘n’-dimensional sphere is very small for large values of ‘n’: almost all ‘space’ lies, as it were, outside the sphere.
In recent years, remarkable results have been achieved, both with classical mathematical work and with the help of artificial intelligence
Determining the kiss number in such spaces is a challenging task, and only a handful of cases have been solved exactly: in dimensions four, eight and 24, the kiss number is 24, 240 and 196,560, respectively. In other dimensions, mathematicians know that certain numbers of kissing spheres are possible because they can construct such configurations. This produces so-called ‘lower limits’ for the coastal number; the actual number of kisses is at least as high.
One application that justifies the investigation of kiss numbers relates to the fact that tangent spheres in higher-dimensional spaces are closely related to correcting errors in communication systems. When transmitting digital signals, bits can shift due to noise. Each possible message is therefore represented as a point in a higher-dimensional space, with a sphere around that point within which errors can still be corrected. The more spheres fit around a central sphere without overlap, the faster the communication.
In recent years, remarkable results have been achieved, both with classical mathematical work and with the help of artificial intelligence. Mathematician Mikhail Ganzhinov did a PhD at Aalto University in Finland and recently took up a postdoc position in Bulgaria. He established three new lower limits: in dimension ten the kiss number is at least 510, in dimension eleven at least 592 and in dimension fourteen at least 1,932. His results were published in October in the trade magazine Linear Algebra and Its Applicationsalthough the underlying research dates from several years earlier: the peer review took a very long time.
Patrick Östergård: “You can fit six coins tightly around one coin. But in three dimensions, twelve spheres do not fit in the same way around a middle sphere.”
Photo Getty Images/iStockphoto
With this work, Ganzhinov brought the kiss number problem under the fifteenth dimension into motion for the first time in twenty years. But while his paper was under review, he was overtaken on dimension eleven by AlphaEvolvea program from the artificial intelligence laboratory DeepMind. In May this year, DeepMind reported that AlphaEvolve had found a configuration in which 593 eleven-dimensional spheres touch a central sphere – one more than Ganzhinov. The actual value is probably even higher. “I think that lower limit of 593 can be increased to well above 600,” says Ganzhinov.
Eleven-dimensional space
To get an idea of how an artificial intelligence arrives at such a result, imagine one central sphere, with radius 1, at a fixed location in eleven-dimensional space. The algorithm is instructed to choose a fixed number of directions in space from the center of that sphere and to place a sphere of radius 1 in each direction, at a distance 2 from the center of the central sphere.
The algorithm then looks at how close those spheres come to each other: for each pair of spheres, the distance between their centers is calculated. If they are closer together than two units, the spheres will overlap, which is not allowed. The configuration then receives ‘penalties’: the more overlap, the greater the penalty. The program tries to get the total number of penalty points as close to zero as possible: while searching and adjusting, it tries to create a valid arrangement of spheres.
Algebra is the core of Ganzhinov’s work, although he also used a computer to get an idea of the underlying structure. Ganzhinov’s supervisor Patric Östergård does not see AlphaEvolve’s minimal improvement – from 592 to 593 in dimension eleven – as a major AI triumph. On the contrary: in other dimensions, AlphaEvolve found no improvements at all. “If this is what a large team of excellent programmers with the best AI software can currently achieve, then it shows that you cannot compensate for a suboptimal general approach with AI, or that AI has not managed to find a significantly better approach,” Östergård emails.
The dimensions in which Ganzhinov and AlphaEvolve found sharper lower bounds for the kissing number seem rather arbitrary: ten, eleven and fourteen. Other mathematicians focused on other dimensions. For example, the duo Henry Cohn Anqi Li: in an article that has not yet been officially published they found improvements in dimensions 17 to 21.
Beautiful structures
Mathematical spaces vary greatly by dimension. A technique that works well in one dimension is unusable in another. There was still much to be gained in the areas examined by Ganzhinov, because no sharp lower limits were known there. Ganzhinov’s approach turned out to work well there. In five dimensions it is more difficult to find a workable method, precisely because the current lower limit of 40 is already very sharp: we know that the kiss number in dimension five is a minimum of 40 and a maximum of 44.
Östergård about the various dimensions: “In the simple two-dimensional world, spheres are just circles on a flat surface. Six coins can be placed tightly around one coin. But in three dimensions, twelve spheres do not fit around a central sphere in the same way. In twenty-eight and twenty-four dimensions we know beautiful structures; there we know the kiss number exactly.”
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