I built an interactive browser lab that places points on a manifold (torus, sphere, cube, arbitrary STL mesh) and optimizes them by maximizing the Shannon entropy of the pairwise-distance distribution rather than doing standard sphere packing.
Whereas the classic Erdős distinct-distances problem asks how many distinct pairwise distances n points must determine, here I treat the multiset of distances as a probability distribution (Gaussian KDE) and maximize its entropy, giving a continuous extremizer in place of the discrete bound.
This, in effect, produces pseudo-attractive and pseudo-repulsive forces that prefer forming filaments and crystal-like structures.
This is mostly just a cool looking experiment; I don't have any claims or findings or a paper.
Runs entirely in-browser with TensorFlow.js — drag to rotate, no install.
https://math.cognotik.com/experiments/geometric-entropy/index.html