In a groundbreaking development, mathematician Maryna Viazovska has made significant strides in solving the sphere packing problem across two complex dimensions: the eight-dimensional and twenty-four-dimensional spaces. This problem explores the wonderful world of the densest packing of identical spheres in n-dimensional space. One that mathematicians and laypeople alike have been captivated by for centuries. Powerful new developments in artificial intelligence (AI) have spurred this research to new heights. They illuminate new and as-yet-unimagined arenas of collaboration between human creativity and machine learning.
Viazovska’s approach to the sphere packing problem was bold. Among many other accomplishments, she was the first to show that the E8 lattice delivers the best packing of spheres in eight dimensions. Of course, it was the application of quasi-modular forms, an extremely rich and powerful mathematical function, that drove her proof. She collaborated with other mathematicians to study what’s called the Leech lattice. In concert, they showed that it is the best possible solution to the sphere packing problem in twenty-four dimensions.
The Sphere Packing Problem
The sphere packing problem poses a fundamental question: how can identical circles or spheres be arranged in a given space to occupy the maximum possible volume? The challenge varies across dimensions. In two dimensions, the nature of the honeycomb structure makes it the most widely known optimal solution. In contrast, in three dimensions, pyramidal stacking of spheres turns out to be the best arrangement.
In higher dimensions, though, the situation is much worse. In particular, Sidharth Hariharan and his co-authors had already done a lot of the heavy lifting in formalizing the 8-dimensional sphere packing proof. They all began sharing their current blueprints with Viazovska. This intervention was key to moving the mathematical dialogue forward on this difficult, technical problem.
“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
The Role of AI in Formal Proof Verification
With the recent development of large AI systems, this space has changed very quickly. A particularly interesting example is Gauss, a cutting-edge AI created to automatically formalize proof by deep learning. In an impressive demonstration of its capabilities, Gauss completed the 8-dimensional sphere packing proof and even identified a typo in an earlier published paper within just five days.
Fostering this success, Math, Inc. introduced a new iteration of Gauss to address deeper, more rigorous challenges. This new and improved autoformalization process managed to autoformalize Viazovska’s 24-dimensional sphere packing proof in just two weeks! The formalization process While brilliant in its depth, the formalization itself was over 200,000 lines of code, accounting for the complex nature of the proof itself.
“One of them helped us identify a typo in our project, which we then fixed,” – Hariharan
Jesse Han, the CEO and co-founder of Math, Inc., calls Gauss a “reasoning agent.” This rare distinction highlights its power to combine grounded, traditional natural language reasoning with state of the art fully formalized reasoning. This unique mix is key to addressing multi-faceted mathematical conjectures.
“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
Collaboration Between Humans and AI
This successful verification of these complex proofs is a momentous achievement in mathematics. This interaction between human mathematicians and machines ai represents the fastest evolutions made by technology within the research industry. Both Hariharan and Han stress how critical this relationship is to making big impact wins.
“This is technology that we’re very excited about, because it has the capability to do great things and to assist mathematicians in remarkable ways,” stated Hariharan. The collaboration didn’t stop at simply formalizing proofs. It strongly filled in gaps in underlying knowledge, which is very important when developing understanding of advanced mathematical concepts.
Han noted the intricacies involved in formalizing the 24-dimensional proof: “It was actually significantly more involved than the 8-dimensional case because there was a lot of missing background material that had to be brought online surrounding many of the properties of the Leech lattice, in particular its uniqueness.“
Liam Fowl remarked on the implications of these results: “These new results seem very, very impressive and definitely signal some rapid progress in this direction.”
The Future of Mathematical Research
AI is changing quickly and becoming an important resource for mathematicians. This new reality provides a timely opportunity to reflect on the evolving roles of mathematicians and programmers. Han reflects on this evolution: “A programmer used to be someone who punched holes into cards, but then the act of programming became separated from whatever material substrate was used for recording programs.” This move reflects a broader change, not just in tactics but in how mathematicians are engaging with deeper social problems.