ZAP // FROM-2
For the first time since John Leech in 1976, progress has been made on the intriguing “kissing problem” – which remains full of mysteries.
It is said that, in May 1694, Isaac Newton and the astronomer David Gregory They began to contemplate the nature of the stars, eventually coming across a mathematical puzzle that would last for centuries.
As described by , the conversation had to do with how stars of different sizes orbit a central sun. But what is remembered today is the more general question that inspired the conversation: Given a central sphere, how many identical spheres can be arranged to touch it without overlapping?
In three dimensions, they can be positioned 12 spheres around the central sphere so that each one touches a single point. But this arrangement leaves spaces between the spheres.
Consequently, Would it be possible to squeeze a 13th sphere into this remaining space? The distinguished scientists questioned. Gregory thought so. Newton didn’t think so.
O “kiss” problemas it came to be called, proved to be relevant, for example, for the analysis of atomic structures and the construction of error correction codes. Furthermore, it also became a great mathematical challenge.
It took until 1952 for mathematics to prove that Newton was right: in our three familiar dimensions, the maximum number of “kisses” is 12.
However, this problem can be posed to the spheres of any dimension.
In two dimensions, the answer is clearly six: If we place a coin on a table, we see that, when we place six other coins around it, they fit perfectly.
But… what about in higher dimensions?
Although the problem has already been solved, in dimensions four, eight and 24, where mathematicians have managed to optimally pack spheres into wonderfully symmetrical lattice structures, in all other dimensions, where there is more space between spheres, the problem remains open.
To improve estimates, mathematicians typically follow the same intuition that gave them solutions in dimensions like 8 and 24: they look for ways to arrange the spheres as symmetrically as possible.
In 2022, a Massachusetts Institute of Technology mathematics student named Anqi Li decided to go in search of these strangest structures.
While working on the project, she came up with a deceptively simple idea that has now allowed her and her advisor, Henry Cohnimprove kiss number estimates across a particularly challenging group of dimensions: 17 a 21.
The study, in November arXivproves that the kiss numbers in 17, 18, 19, 20 and 21 dimensions are at least 5.730, 7.654, 11.692, 19.448 e 29.768respectively.
The work marks the first progress on the problem along these dimensions since the 1960s — and shows the benefits of injecting more clutter into potential solutions.
“From codes to kisses”
In 1967, mathematician John Leach He used an incredibly efficient code – famous for being later used by NASA to communicate with the Voyager probes – to build a network of points that he eventually named.
Fifty years later, Cohn and several other mathematicians proved that it is possible to use this lattice to pack spheres as densely as possible in a 24-dimensional space.
At first, Professor Cohn was skeptical. It’s easy to make a small mistake in this type of calculation, especially when it comes to positive thinking. To “disprove” the theory, they checked their new arrangement of points on a computer. And it worked: All spheres fit correctly.
That year, Li went to work with Cohn as intern at Microsoft Researchwhere they could carefully refine the error-correcting codes they were using so they could continue adding compatible spheres to Li’s “weird” 17-dimensional structure.
In the end, they managed add 384 new spheres to the 1967 Leech-based estimate, raising the lower limit on the number of kisses to 5.730.
They then applied similar techniques to improve the number of kisses in dimensions 18 to 21.
And after 21?
However, in dimensions 22 and 23, the strategy failed. It appeared the pair had exhausted their sign-flipping approach.
Despite this, mathematicians became more interested in how Cohn and Li achieved their gains.
Their new structures are very different from the highly symmetric structures inspired by the Leech network. The code-based methods they used to add spheres gave them more irregular configurations – something entirely new.
Several recent results support the promise of these less accessible possibilities. As Quanta Magazine reports, in the last two years, mathematicians have created new intelligent constructions in dimensions 5, 10 and 11, bending or breaking the usual rules of symmetry.
“Each unusual structure that is discovered gives little hints and clues to the truth. The kissing problem is still full of mysteries“, considered Henry Cohn, quoted by the same magazine.
