Human ingenuity outpaces AI in finding new 'kissing number' bounds
The age-old question of how many coins can touch one coin or how many basketballs can "kiss" one basketball at the same time has captivated mathematicians for centuries. This riddle, known as the kissing number problem, becomes increasingly complex as the dimensions increase beyond 4D. Despite its playful name, the problem has practical applications in mobile communications and satellite navigation.
Aalto University doctoral candidate Mikhail Ganzhinov has made significant strides in this field, establishing three new lower bounds for the kissing number: at least 510 in dimension 10, at least 592 in dimension 11, and at least 1,932 in dimension 14. His findings were published in the journal Linear Algebra and its Applications.
For two decades, progress on the riddle had stagnated for dimensions below 16 until earlier this year when AlphaEvolve, an AI developed by Google's DeepMind laboratory, made headlines. It increased the lower bound for dimension 11 to 593, just one step behind Ganzhinov's result. So, how did the researcher surpass the AI in dimensions 10 and 14?
Ganzhinov explains, "I reduced the problem size by focusing on highly symmetric arrangements. The current lower bound for dimension 11 is still quite weak, and I believe it can be pushed well beyond 600."
His thesis advisor, Professor Patric Östergård, acknowledges the impressive outcome and highlights the limits of AI. "Artificial intelligence can achieve remarkable feats, but it's not omnipotent. The game may still be in Mikhail's favor in Dimension 11, too," he remarks.
Despite his recent PhD, Ganzhinov remains humble, recognizing the rapid evolution of the field. Professor Henry Cohn from MIT and researcher Anqi Li are set to publish new results that extend the kissing number bounds in dimensions 17 to 21, marking the first progress in those dimensions in over 50 years. Ganzhinov believes his findings are part of a broader wave of recent developments.
"This riddle has challenged mathematicians since the famous conversation between Newton and Gregory," says Ganzhinov. "Yet solving it also has a practical purpose: understanding connections to spherical codes has real-life implications in the field of communications."
The full story continues: Mikhail Ganzhinov, Highly symmetric lines, Linear Algebra and its Applications (2025). DOI: 10.1016/j.laa.2025.05.002 (https://dx.doi.org/10.1016/j.laa.2025.05.002)
