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?
An equally valuable form of support is to simply share the videos.

The artwork in this video is by Kurt Bruns, aided by Midjourney

Here are several other good posts about the classic Poisson proof

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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These animations are largely made using a custom python library, manim. See the FAQ comments here:

You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

Download the music on Bandcamp:

Stream the music on Spotify:

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Various social media stuffs:

[billtruttschel]

Grant, as a lowly college lecturer with insufficient funds to donate to your cause, I must nonetheless congratulate you on another masterpiece. Your visualizations are second to none and your teaching is beyond fantastic. Thank you for your contributions to mathematics.

[DasIllu]

I think pi gets really sad whenever e is not around. It's not just love. It's a rather complex relationship. No wonder, both of them seem to be a bit irrational. Especially e gets fouriously impotent when pi is not around, despite pi's negativity.

[kylehart]

"Who ordered another dimension" 😂 classic mathematician path to solving a problem

[dominicveconi]

Grant, I’m a mathematician and math educator. I’ve of course seen the integration proof of the computation of the area under e^{-x^2}, but never in my life have I either seen or come up with such an elegant demonstration for why we MUST expect pi to show up in the Gaussian. Thank you, sir. Actually this gives me an idea for an in-class activity for my future analysis students…

[minerharry]

1:45 put such a wide smile on my face and reminded me why I love this channel so much. So often the explanation for why a thing is is entirely proof based. I love proofs, and coming up with proofs is a wonderful experience of problem solving, but on their own they cannot *satisfy* my disbelief. Stuff like this, using the concepts and *reasoning* /within/ the proofs to make a point, is exactly what I love about math. Thank you, grant!

[aucsiya]

Just watched the video again with your Korean AI voice. As a Korean, I'm genuinely blown away—it sounds incredibly natural! Imagining how this will broaden accessibility to your fantastic math content is truly exciting 👏

[jmcsquared]

That we live in an age where educators have the opportunity to unpack the meaning and history behind some of the greatest mathematical discoveries for a substantially large audience is a privilege that we should all be infinitely grateful for.

[histeward]

The Korean version sounds very natural. The pipeline works incredibly well!

[fari]

Him saying "feel less out of the blue" at 22:10 after deriving the proof visually and in BLUE is like the 10th bonus point for this channel. I love it so much

[moralboundaries]

I love everything about your videos. Your amazing animations, masterful scripts, pleasant and well recorded voice, tight editing. It all comes together to create some of the best educational content the world has ever seen. Thank you for sharing this for free and enriching the intellectual lives of so many people. ❤

[paradoxicallyexcellent]

More than perhaps any video in 3b1b, this one shows how learning math history makes one a better mathematician. What a great lesson!

[imsayif]

Who else came here just to listen Grant speak Korean?

[coffeeguy.]

Finally, the much awaited 3b1b statistics series is on a roll!

[-bj]

This makes me think: a series on statistics would be excellent. I am sure there is a lot of visualisation behind all the sum of squares and F statistics, of interactions and everything, that are never taught. Even "serious" books barely talk about the intuition of the sum of squares beyond how they are derived from LRT.

[ahmedgabr]

I doubt anyone can possibly make a better visualization for explaining this proof. The quality of your videos is truly on another level

[johnchessant]

Even though I've seen this proof like a hundred times, it still brings a smile each time especially when animated as beautifully as here! And 3b1b did not disappoint, this really is a new perspective on it and I can't wait for the next video.

[LillianRyanUhl]

You know I actually learned the proof about spheres you mentioned by hand proving it for my harmonic analysis class last semester! I was hoping to be able to learn more about abstract harmonic analysis and/or wavelets when I set out in that course, but it actually wound up being a very classical class, and featured many diversions on historical analytic number theory, special functions, and computational techniques. It was not at all what I was expecting at the outset, but it was fascinating!

Also, I'll mention that I was one of the people who got to meet you at JMM, and got a picture! I've been following your content for years, and you're still such an inspiration to the way that I communicate math to non-mathematicians, even if I know most of the content that you post these days! I still find it very interesting and always a good review though :)

[andrerenault]

I love the pieces of art that aren’t normally part of the “aesthetic” of 3b1b but somehow still fit right in.

[ArgentineTangoRadio]

Great explanation, as usual! Minor historical correction: at 21:40 you say that Maxwell "independently stumbled upon the same derivation" as Herschel. The current scholarly consensus is that Maxwell read Herschel's paper and adapted his proof to the kinetic theory of gases, so it was not independently discovered. For details (as well as details that may be useful for your next promised video) see B. Gyenis (2017) "Maxwell and the normal distribution" in Studies in History and Philosophy of Modern Physics vol 57, doi: 10.1016/j.shpsb.2017.01.001 .

[mayurmatada]

Just wanna Say, Thanks for doing whatever you are doing. Never stop 3B1B