Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r\in [0,1]$ so there an inscribed rectangle in $\gamma$ aspect ratio $\mathrm{tan}(r\cdot \pi/4)$, then Lebesgue measure at least $1/3$. To do this, we study sets disjoint homologically nontrivial projective planes smoothly embedded $\mathbb{R}\times \mathbb{R}P^3$. any such can be equipped with a natural total ordering. combine this ordering Kemperman's theorem $S^1$ to $1/3$ sharp lower bound on probability Möbius strip filling $(2,1)$-torus knot solid torus times interval will intersect its rotation by uniformly random angle.