RANDOM TRIANGLES ON FLAT TORI

Olivier Glorieux, Andrew Yarmola

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)1345-1354
Number of pages10
JournalRocky Mountain Journal of Mathematics
Volume52
Issue number4
DOIs
StatePublished - Aug 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • geodesic
  • random
  • tori
  • triangles

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