Why π is in the normal distribution (beyond integral tricks)

Where's the circle? And how does it relate to where e^(-x^2) comes from?Help fund future projects:   / 3blue1brown  
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The artwork in this video is by Kurt Bruns, who used Midjourney as part of the process for some pieces. I wrote more about the specifics of that process and my thoughts around it here:   / thoughts-on-ai-83881719  

Here are several other good posts about the classic Poisson proof

vcubingx:    • Why does pi show up here? | The Gauss...  
BriTheMathGuy:    • This isn't a Circle - Why is Pi here?  
Dr. Alter's math library: https://idan-alter.github.io/2023/02/...

And if you'd like to see many other variations on approaching this integral, take a look at this expository paper from Keith Conrad: https://kconrad.math.uconn.edu/blurbs...

Correction: 13:46 - The denominator should read 2πσ^2

Timestamps:
0:00 - The statistician's friend
3:44 - The classic proof
12:47 - The Herschel-Maxwell derivation
21:55 - Reflecting back on the proof
23:50 - A bonus problem

Thanks to these viewers for their contributions to translations
French: Jeremy Senn, Laurent Joubert
German: dreivmeister, lprecord, mathiashaebich
Hebrew: David Bar-On, Omer Tuchfeld
Hindi: prateekbansal97
Spanish: Joel Tovar
Turkish: kilobravo3

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https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/

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Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/

Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...

Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...

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