Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to √(3)

2018/03/16 by Hugelmeyer, Cole
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1803.07417

Abstract

We use Batson's lower bound on the nonorientable slice genus of (2n,2n-1)-torus knots to prove that for any n ≥ 2, every smooth Jordan curve has an inscribed rectangle of of aspect ratio tan((πk)/(2n)) for some k∈ \1,...,n-1\. Setting n = 3, we have that every smooth Jordan curve has an inscribed rectangle of aspect ratio √(3).

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