Reproduce “Drums of high width” on OpenResearch

About this paper

Alex Davies, Prateek Gupta, Sebastien Racaniere, Grzegorz Swirszcz, Adam Zsolt Wagner, Theophane Weber, Geordie Williamson

Researchers at Google DeepMind constructed a family of 5-dimensional drums whose excess width grows linearly with the number of vertices, addressing an open problem in combinatorial geometry. This work yields a new infinite family of high-dimensional polytopes that violate the Hirsch conjecture, achieving the largest known Hirsch excess, approaching 1/16.

Key insights

• Constructing complex polytopes by leveraging underlying symmetry groups can be an effective strategy for efficiently searching for specific properties, like increased diameter.
• The "double pancake" geometric structure, characterized by significantly different coordinate scales, was identified computationally as a key motif correlating with higher-width drums.
• The pair embedding map, crucial for width bounds, maps to specific "boundary edges" of the drum skins, simplifying the analysis of incidence patterns between drum skins.

Summary by alphaXiv.