General-purpose math software just solved impossible geometric puzzles that used to require custom-built supercomputers.
Off-the-shelf global optimization solvers found new best-known solutions for packing circles and polygons just by describing the rules more effectively. This suggests that general math software has reached a tipping point where it can outperform specialized, handcrafted algorithms. Researchers no longer need to build custom solvers for every new non-convex problem they encounter. By simply using the right mathematical formulation, standard tools can now crack some of the hardest puzzles in geometry. This democratizes high-end optimization for engineers and designers across every industry.
Out-of-the-Box Global Optimization for Packing Problems: New Models and Improved Solutions
arXiv · 2605.04850
Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct nonlinear formulations, we consider packing circles in squares and fixed-perimeter rectangles, packing circles into minimum-area ellipses, packing regular polygons into regular polygons, and packing Platonic solids into Platonic solids. For ellipse packing, w