Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, and a clever topological solution to a weaker version of the question. Brought to you by you: http://3b1b.co/topology-thanks 
This video is based on a proof from H. Vaughan, 1977.
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[youtube https://www.youtube.com/watch?v=AmgkSdhK4K8]

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