The inscribed square/rectangle problem, solved using Möbius strips and Klein bottles.
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This argument was originally by Herbert Vaughan, appearing for examples in this issue of the Topology Proceedings.
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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:
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Nice Quanta article about this result:
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Timestamps:
0:00 - Inscribed squares
1:00 - Preface to the second edition
3:04 - The main surface
10:47 - The secret surface
16:45 - Proving collision
22:38 - Why are squares harder?
25:10 - What is topology?

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These animations are largely made using a custom Python library, manim. See the FAQ comments here:
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All code for specific videos is visible here:
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The music is by Vincent Rubinetti.
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3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly.

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