One of the most elegant demonstrations in topology is the proof of the *inscribed rectangle problem* (a solved variant of the unsolved *inscribed square problem*) which states that for any plain, closed continuous closed curve $\Gamma$ in $\mathbb{R}^2$, there exist four points that are the corners of an inscribed rectangle. (There are variants of this problem involving cyclicality as well.)

The proof relies on a clever representation of the locations of unordered pairs of points on $\Gamma$ with a point $p$ on a Möbius strip, and a function $f(p)$ that represents the Euclidean distance between the points on $\Gamma$.

The key step in the proof invokes the topological fact that the mapping of the Möbius strip and $f(p)$ to the plane of $\Gamma$, involving the unwrapping of the strip's boundary to coincide with $\Gamma$ guarantees that there exist two points, $p_1$ and $p_2$ that map to the same point on the plane *and* have the same value, i.e., $f(p_1) = f(p_2)$.

Is there a good reference, proof, or even intuitive demonstration of this fact?