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arXiv · 2003.08190

Random triangles on flat tori

Olivier Glorieux·Andrew Yarmola
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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 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.

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Olivier Glorieux, Andrew Yarmola. 2020-03-18. Random triangles on flat tori. https://arxiv.org/abs/2003.08190

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