In a remarkable breakthrough that defies the current limits of artificial intelligence, a doctoral candidate at Aalto University has pushed the boundaries of a long-standing mathematical puzzle known as the kissing number problem. This iconic question, which asks how many non-overlapping spheres can simultaneously touch a central sphere, has stymied mathematicians for centuries, especially in higher-dimensional spaces. The problem’s complexity escalates dramatically as dimensions increase, making it a prominent challenge in both pure mathematics and applied fields like telecommunications and satellite navigation.
Mikhail Ganzhinov, a doctoral researcher under the mentorship of Professor Patric Östergård, recently announced new lower bounds for the kissing number in dimensions 10, 11, and 14. His results demonstrate a substantial leap in understanding: at least 510 spheres can kiss a center sphere in 10 dimensions, 592 in 11 dimensions, and an astonishing 1,932 in 14 dimensions. Notably, these new bounds represent the first significant progress in these dimensions for over two decades, a period during which the problem seemed unsolvable.
Ganzhinov’s success is especially striking given the rise of AI-powered approaches. In May, the AI system AlphaEvolve, developed by DeepMind, made headlines by improving the kissing number lower bound in the 11th dimension to 593, surpassing previous human efforts. However, in dimensions 10 and 14, the human researcher outperformed this cutting-edge AI. This juxtaposition highlights a profound insight: despite rapid advances in machine learning and computational intelligence, human intuition and innovative methodological design remain indispensable in solving complex mathematical conundrums.
The key to Ganzhinov’s approach lies in his strategic reduction of the problem’s scope by focusing on highly symmetrical configurations. Symmetry, a fundamental concept in mathematics, allows for the simplification of otherwise intractable problems by exploiting repetitive patterns and structural regularities. By narrowing the search for kissing arrangements to those manifesting a high degree of symmetry, Ganzhinov efficiently trimmed down computational complexity while preserving mathematically significant solutions.
This focus on symmetry is not merely a clever trick; it aligns closely with the natural tendencies of high-dimensional geometric configurations. Symmetric arrangements often correspond to optimal or near-optimal packings in multiple dimensions, making them fertile ground for discovering new kissing numbers. Ganzhinov’s method thus melds deep theoretical insight with practical computational techniques, showcasing the intricate interplay between abstract mathematics and modern algorithm design.
Professor Östergård, Ganzhinov’s thesis advisor, notes that the results underscore important limitations in present AI capabilities. “Artificial intelligence can accomplish extraordinary feats,” he reflects, “but it is far from omnipotent. There remain areas where human creativity and mathematical intuition hold the edge, at least for now.” The professor’s words encapsulate a broader discourse in the scientific community about the evolving roles of human researchers and machines in pushing the frontiers of knowledge.
Beyond the immediate mathematical interest, the kissing number problem has significant implications for applied sciences, particularly in communication technology. The arrangement of spheres in high-dimensional spaces relates to spherical codes, which underpin error-correcting codes and signal transmission in noisy environments. Improved bounds on kissing numbers can lead to denser packing of signals, thereby enhancing data throughput and reliability in systems ranging from mobile networks to satellite communications.
The historical roots of the kissing number problem run deep. The puzzle famously emerged from an exchange between Sir Isaac Newton and David Gregory in the 17th century, focusing initially on three-dimensional spheres. Extending intuitive spatial notions into higher dimensions rapidly escalates difficulty, rendering exact solutions rare and prized achievements. The problem’s longevity and resistance to solution underscore its foundational role in discrete geometry and number theory.
Recently, the momentum in kissing number research has accelerated. Alongside Ganzhinov’s findings, other prominent mathematicians, including Professor Henry Cohn of MIT and researcher Anqi Li, are producing advances extending the problem’s scope in dimensions 17 to 21. These fresh breakthroughs promise to rejuvenate a field that, for decades, had seen relatively little progress. This renewed activity signals a vibrant era in geometrical and combinatorial mathematics, propelled by a blend of computational power and novel theoretical insights.
Ganzhinov approaches his accomplishment with a measured humility, aware of the rapid evolution of the field he contributes to. His work forms part of an ongoing wave of discoveries redefining the modern boundaries of mathematical knowledge. He emphasizes that while the kissing number represents a classical problem, the methods and outcomes resonate with current technological challenges, particularly in signal theory and the geometry of information.
This hybrid narrative of mathematical tradition and cutting-edge innovation exemplifies the evolving landscape of research today. As computational methods grow in sophistication, the interplay between artificial intelligence and human ingenuity becomes increasingly complex and collaborative. Ganzhinov’s work provides a case study in how focusing on problem structure—here, symmetry—can yield breakthroughs even in the face of formidable algorithmic competition.
In sum, the recent strides in supporting kissing number bounds reflect much more than abstract numerical advances. They signify progress that blends centuries-old mathematical heritage with tomorrow’s technological imperatives. Ganzhinov’s results illuminate pathways not only for pure geometric understanding but also for practical enhancements in communication frameworks that permeate our interconnected modern world.
The dialogue between human reasoning and artificial intelligence, embodied in this research, points toward a future of mutual reinforcement rather than outright competition. As researchers continue to explore high-dimensional geometry, the kissing number problem remains a vibrant testament to the challenges and triumphs possible when tradition meets innovation head-on.
Subject of Research: Kissing Number Problem in High-Dimensional Geometry and Its Applications in Communications
Article Title: Highly Symmetric Lines
News Publication Date: 1-Oct-2025
Web References:
- Mikhail Ganzhinov’s article: https://www.sciencedirect.com/science/article/pii/S0024379525001946?via%3Dihub
- Related new results by Henry Cohn and Anqi Li: https://www.arxiv.org/abs/2411.04916
References:
- Mikhail Ganzhinov, Highly Symmetric Lines, Linear Algebra and its Applications, DOI: 10.1016/j.laa.2025.05.002
Image Credits: Kira Vesikko / Aalto University
Keywords: Kissing number, high-dimensional geometry, sphere packing, symmetry, algorithmic mathematics, artificial intelligence, signal processing, spherical codes, satellite navigation, mobile communications
