Abstract
Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to a new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.
Citation
Olivier Glorieux. Andrew Yarmola. "Random triangles on flat tori." Rocky Mountain J. Math. 52 (4) 1345 - 1354, August 2022. https://doi.org/10.1216/rmj.2022.52.1345
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