Advancing Geometry with AI: Multi-agent Generation of Polytopes

TL;DR

This paper presents an AI system that autonomously generates complex polytopes, advancing geometric research by producing novel extremal examples in challenging problems like the Hirsch Conjecture, surpassing human-crafted bounds.

Contribution

The authors introduce a highly parallel AI system capable of generating complex polytopes, demonstrating its effectiveness on longstanding geometric problems and surpassing previous human achievements.

Findings

AI generated polytopes surpass previous bounds in the Hirsch Conjecture

System produced millions of examples with novel properties

Automated generation accelerates discovery in complex geometric problems

Abstract

Polytopes are one of the most primitive concepts underlying geometry. Discovery and study of polytopes with complex structures provides a means of advancing scientific knowledge. Construction of polytopes with specific extremal structure is very difficult and time-consuming. Having an automated tool for the generation of such extremal examples is therefore of great value. We present an Artificial Intelligence system capable of generating novel polytopes with very high complexity, whose abilities we demonstrate in three different and challenging scenarios: the Hirsch Conjecture, the k-neighbourly problem and the longest monotone paths problem. For each of these three problems the system was able to generate novel examples, which match or surpass the best previously known bounds. Our main focus was the Hirsch Conjecture, which had remained an open problem for over 50 years. The highly parallel A.I. system presented in this paper was able to generate millions of examples, with many of them surpassing best known previous results and possessing properties not present in the earlier human-constructed examples. For comparison, it took leading human experts over 50 years to handcraft the first example of a polytope exceeding the bound conjectured by Hirsch, and in the decade since humans were able to construct only a scarce few families of such counterexample polytopes. With the adoption of computer-aided methods, the creation of new examples of mathematical objects stops being a domain reserved only for human expertise. Advances in A.I. provide mathematicians with yet another powerful tool in advancing mathematical knowledge. The results presented demonstrate that A.I. is capable of addressing problems in geometry recognized as extremely hard, and also to produce extremal examples different in nature from the ones constructed by humans.