The inscribed square/rectangle problem, solved using Möbius strips and Klein bottles.
Playlist with more neat proofs:
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This argument was originally by Herbert Vaughan, appearing for examples in this issue of the Topology Proceedings.
The on-screen argument for why all closed non-orientable surfaces must intersect themselves in 3d is a slight variation on one I heard from Dan Asimov.
2020 Paper by Greene and Lobb:
Nice Quanta article about this result:
https:// ...
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