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arXiv · 1803.07417

Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to $\sqrt{3}$

Cole Hugelmeyer
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Abstract

We use Batson's lower bound on the nonorientable slice genus of $(2n,2n-1)$-torus knots to prove that for any $n \geq 2$, every smooth Jordan curve has an inscribed rectangle of of aspect ratio $\tan(\frac{\pi k}{2n})$ for some $k\in \{1,...,n-1\}$. Setting $n = 3$, we have that every smooth Jordan curve has an inscribed rectangle of aspect ratio $\sqrt{3}$.

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Cole Hugelmeyer. 2018-03-16. Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to $\sqrt{3}$. https://arxiv.org/abs/1803.07417

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