A beautiful solution to the inscribed rectangle problem.
Playlist with more neat proofs: https://www.youtube.com/playlist?list=PLZHQObOWTQDPSKntUcMArGheySM4gL7wS />Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support
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This argument was originally by Herbert Vaughan, appearing for examples in this issue of the Topology Proceedings.
https://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf

2020 Paper by Greene and Lobb:
https://arxiv.org/pdf/2005.09193

Nice Quanta article about this result:
https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/

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:
https://3b1b.co/faq#manim
https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/

All code for specific videos is visible here:
https://github.com/3b1b/videos/

The music is by Vincent Rubinetti.
https://www.vincentrubinetti.com
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

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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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