Scientists at Fudan University, Peking University, and the Shanghai Academy of AI for Science (SAIS) used an artificial intelligence system called Packing Star to solve a 300-year-old maths problem posed by none other than Isaac Newton.
Known as the kissing number problem, it has been solved for common numbers of dimensions, such as the three-dimensional world we live in. But answers to the problem in other dimensions remained elusive.
Finding them now opens up avenues in large-scale data storage and advanced telecommunications, Chinese media reported.
What is the kissing number problem?
In 1694, British polymath Isaac Newton engaged in a debate with Scottish mathematician David Gregory about how many spheres a central sphere of the same size could touch or ‘kiss’ without overlapping other spheres. This is the kissing number problem, one of the 23 sets of problems in the field of mathematics.
In the world we live in, this is akin to a pomegranate packed with seeds, and the question is how many seeds can a central seed potentially kiss, if it were shaped like a sphere. Newton and Gregory debated about the answer, with the former arriving at the number 12, while the Scotsman calculated the number to be 13.
More than two centuries later, we had the definite answer of 12, proving that Newton was correct. But the solution is only true for three dimensions. As the number of dimensions changes, the number of spheres that the central sphere can kiss changes.
Mathematician Oleg Musin demonstrated in 2003 that the answer to the problem in four dimensions was 24, while in the 1970s, the answer in 24th dimension was 196,950. Yet, the solutions for a larger number of dimensions remain elusive.
Leveraging AI for solutions
A collaborative effort by scientists at premier institutes in China led to the development of a reinforcement learning system called PackingStar to solve the kissing problem in higher dimensions.
Inside PackingStar, two AI agents work cooperatively to find high-dimensional spaces that are too complex for standard computers to process. The AI is trained from scratch without human input and searches for high-dimensional kissing configurations with clear mathematical structures.
Leveraging AI for this problem is not just part of the AI hype that surrounds it, but also a necessity for technological advancement. “As the dimension increases, human understanding becomes more limited due to the growing geometric complexity of high-dimensional spaces,” the scientists wrote in their pre-print paper on arXiv.
However, knowing the answers to the problem in different dimensions is not just for mathematical purposes. It has real-world applications, too. For instance, the solutions can help compress information into the fewest bits or determine optimal distributions of communication signals for use in satellite communication or quantum coding.
Using the AI, the team constructed sphere arrangements in dimension 13 and identified thousands of arrangements that could advance understanding of sphere packaging.
The team, however, told South China Morning Post that the AI does not provide mathematical proof of these new configurations, and humans need to verify the results.