The sphere packing problem has captivated mathematicians for centuries. They want to find out how closely the same spheres can be packed in various dimensions. In the past few years, exciting breakthroughs have been made in this field. Mathematician Maryna Viazovska and those upon whom she depended to contribute were instrumental in these recent advancements. They proved the sphere packing problem in 8 dimensions, known as the Kepler conjecture, and in 24 dimensions. This remarkable feat resulted in stunning discoveries, now long made official with the assistance of cutting-edge artificial intelligence.
Starting in March 2024, the team kicked off the “Formalising Sphere Packing in Lean” project. Specifically, though, they want to convert Viazovska’s proofs into Lean — a domain-specific programming language that’s increasingly popular among mathematicians for writing and checking proofs. Mathematicians and AI assistants such as Gauss have partnered in a thrilling collaboration. This partnership is a great example of the remarkable power of AI to address complicated mathematical verification challenges.
The Sphere Packing Problem
The sphere packing problem raises an essential question: how densely can identical circles and spheres be arranged in n-dimensional space? To the best of our knowledge, the theoretical solution is only known in two dimensions where the optimal arrangement is the honeycomb tessellation. As dimensionality increases, the challenge of the task increases dramatically.
Similarly, Viazovska showed that the E8 lattice indeed gives the densest packing in 8-dimensional space. Her proof relied on deep mathematical functions known as quasi-modular forms. This method significantly deepened our comprehension of geometry in higher dimensions. In addition to this, she, with her co-authors, proved that the Leech lattice is the optimal packing arrangement in 24 dimensions.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl.
This blending of world-leading theoretical development with real-world execution is an example of the changing way that mathematicians are tackling today’s big challenges. The findings not only deepen mathematical understanding, but pave the way for further investigation in higher dimensional spaces.
The Role of Artificial Intelligence
In a time when technology is ever more blending with STEM fields, AI has been key in formalizing mathematics proofs. Gauss, a new AI system developed by Math, Inc., achieved amazing feats at record speed. It managed to autoformalize Viazovska’s 24-dimensional sphere packing proof in under two weeks. While that was a great accomplishment, the greater impact of this success was showing what AI-human collaboration intended to scale through autoformalization could achieve.
When a formalization process for the 8D proof was announced on February 23, only five days later, Gauss found and fixed a mistake — a typo that would have rendered the whole published paper incorrect. This kind of power and efficiency is a great example of how AI can support and enhance human work here in research and validation.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan.
Harian and his collaborators recognized that while Gauss conducted an automated effort, many human contributions laid the groundwork for this success. AI’s incorporation into the verification process represents a true paradigm shift in the world of mathematical research.
The Future of Mathematical Proof Verification
As the landscape of mathematical research continues to evolve and change, the role of mathematical programming languages like Lean adapts. As a proof assistant, Lean allows mathematicians to write their proofs in the Lean language, with absolute guarantees of correctness provided by computer verification. Through this collaborative and creative approach, mathematicians are becoming better equipped to work with advanced AI systems.
Jesse Han, CEO and co-founder of Math, Inc., calls Gauss a reasoning agent. It lies in-between a very traditional natural language reasoning and fully formalized reasoning. This combined approach provides for a deeper investigation into abstract, sophisticated math ideas.
“It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural language reasoning and fully formalized reasoning,” – Jesse Han.
The partnership between AI and human researchers is not limited by replacing humans with machines. It represents a remarkable breakthrough in the way we formalize and check mathematical proofs, allowing more rapid progress to be made in our comprehension of mathematics.