Is this the most beautiful proof? (Fermat's Two Squares)

A visual explanation of one-sentence proof of Fermat's Two Squares Theorem by Don Zagier.
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This video adapted a lot of material from this paper: https://arxiv.org/abs/2112.02556
If you're interested in this proof, I highly recommend it and Proofs From The Book! (In general, if you're interested in math, you should read Proofs From The Book)

Some Clarifications:
At 9:00, I meant to say 3^2 + 2^2. Sorry :p
When it comes to the Zagier Map, one question a friend asked is that we only ever consider cases where the inner square is the smallest square or the largest square - what about the in between cases? To this, try and think of if you can create a windmill with an "in-between" case. If you can/cannot, why? (spoiler you can't!)

The source code for the animations can be found here:
https://github.com/vivek3141/videos

A huge thanks to Chaeyeon Lee, Youngjin Park, Oliver Ni, Eric Che and Antonio Kam for helping me review the video!

These animation in this video was made using 3blue1brown's library, manim:
https://github.com/3b1b/manim

Chapters:
0:00 Introduction
1:40 Part 1: The Involution
5:42 Part 2: The Windmill
10:47 Part 3: The Zagier Map
15:12 Putting it all together
17:13 Outro

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Music (in order):
Philanthrope, mommy - embrace https://chll.to/7e941f72
GameChops - National Park
Philanthrope, Idealism - Still https://chll.to/a110849c
GameChops - Route 113
Helynt - Route 47
Helynt - Bo-omb Battlefield
Helynt - Undewater
Knowmadic - Faces https://chll.to/892bc12e
Helynt - Littleroot Town

Hope you guys enjoyed the music! I especially had fun picking out the tracks for this video. S/o to Helynt, they make amazing remixes!

The most beautiful proof in math? (Fermat's Two Squares Theorem)

Some tags: number theory, fermat's two squares, windmills, proofs, mathematics, vcubingx v cubingx, vcubing x