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