This open problem taught me what topology is

This episode explores the inscribed square problem, a mathematical puzzle questioning whether every closed continuous loop contains an inscribed square. Against this backdrop, the speaker delves into a simpler, solved version: proving every closed loop contains an inscribed rectangle. The solution involves a clever mapping of loop point pairs to a three-dimensional surface, where self-intersection of this surface corresponds to inscribed rectangles. More significantly, the analysis reveals the unexpected relevance of topological concepts like Mobius strips and Klein bottles as problem-solving tools. For instance, the space of unordered pairs of points on a loop is shown to be a Mobius strip, and the impossibility of embedding a Mobius strip in 3D without self-intersection under certain constraints leads to the proof. The episode concludes by discussing the unsolved inscribed square problem and its connection to higher-dimensional embeddings and the concept of smoothness in curves, highlighting the power of topology in solving seemingly unrelated geometric problems.