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
Special thanks to these supporters: www.3blue1brown.com/lessons/g...
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
vcubingx: • Why does pi show up he...
BriTheMathGuy: • This isn't a Circle - ...
Dr. Alter's math library: 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: 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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These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@3blue1brown]

The follow-up to explain how this fits into the central limit theorem: kzbin.info/www/bejne/mpDUp396ndCaZpI That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: kzbin.info/www/bejne/f5K2eKSHlpqCY68

[@apointonacurve]

Still, why is the 1/2 in the exponent?

[@redsix5165]

5:08 just some feedback- I dont think you should display the integral expressed as a function of time and not explain that. Too often y’all are just too smart and drop big ideas like they are obvious. It is not obvious how your function can be expressed as x then swap out x for t. Maybe its just poor “math grammar” that science has adopted…idk - bc philosophically- if you need the integral function to be expressed as time (?acceleration) and the function is purely based on space, that is quite interesting

[@Voice-OverM]

I can help you my friend I work as a voiceover ...I am arabian ...I can do voiceover. I will present it to you "free" as a thank you gift, in support of the channel

[@kylehart643]

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

[@InTrancedState]

Only PhD mathematicians have enough math money to order the 11 dimensions needed for string theory

[@agrajyadav2951]

​@@InTrancedState We only get 3 and these elites get 11 Unfair

[@gallium-gonzollium]

“Sir, this is a Wendy’s”

[@joshuamark1601]

I'll have 4 dimensions with extra dip.

[@numairsayed9928]

String theorists af

[@sayefhasan9339]

Who else came here just to listen Grant speak Korean?

[@maxpettit2343]

Me!!

[@afnankabir2190]

That'd be me, even tho it not being my native language

[@florianadamczyk8208]

Yep ;)

[@karl70552]

Me (but I also wanna know why pi shows up in the normal distribution)

[@knitd1985]

🙋‍♂️

[@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.

[@oszkarvarnagy7896]

raise this man to the top of the comments.

[@zacwarnest-knowles9139]

@@oszkarvarnagy7896 fax

[@jaw0449]

Same

[@0ptic0p22]

god damn ma man be talking like nepolean

[@JO-gz8om]

@@jaw0449I’ll

[@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.

[@david13579naranja]

I feel like impotent is not a random word but another reference I am missing

​@@david13579naranja potentiation is just another name for "raising a number to the power of another"

[@user-iv4dh7zp7s]

e^(i*pi)+1=0 is probably the most wholesome thing in the mathematical world

[@doodmithut7844]

Their relationship truly transcends the ones of real life.

[@defenestrated23]

Bravo!

[@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!

[@simonvutov7575]

fun times

[@3blue1brown]

I do think the goals of understanding and proof should be thought of as separate. Both are deeply important and sometimes they coincide. It's often incredibly fun and enriching to follow a good proof with the exercise of saying "right then, so where on earth did that come from?"

[@edwardlulofs444]

Well, it certainly makes learning this material easier. From experience, it was much more difficult to learn it only by textbooks, professors and interacting with other students. But grad school was one of my favorite ways to spend time.

[@Houdini111]

Yeah. Using math to solve math does make it far less real feeling. You just have to take it at its face value. Trying to relate these numbers and concepts back to the real world is hard (which is basically the whole point of this channel) but is very satisfying to see.

[@carstenmeyer7786]

@@3blue1brown While I'd generally agree (especially when cumbersome rigorous notation may hide the underlying principles of a proof), very satisfying proofs often seem to combine their concepts and the necessary notation elegantly and effortlessly. It is a bit like witnessing skilled craftsmanship -- using the exact right tool for a job leads to not only a beautiful result, but a very satisfying process to get there in the first place. Thank you very much for capturing that beauty!

[@fari1964]

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

[@huawafabe]

Three parts blue 🔵, one part brown 🟤

[@publiconions6313]

Really .. the best source of the feeling of epiphany this channel is. People love to criticize the internet.. but 30 years ago, you would've had to pay a bunch of $ to aquire such regular epiphanies

[@pigismyname]

Does your bully call you blue butt too?

[@cod3builder701]

what are the other 9

[@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 👏

[@lifinale]

I should inform you that this is AI lol

[@yoonhenri4713]

@@lifinaleWOW I think he had no idea when he wrote that

[@Karlswebb]

@@lifinaleYes… they said his AI voice…

[@lifinale]

@@Karlswebb they edited the post…

[@jmcsquared18]

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.

[@dominicveconi4301]

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…

[@hierroflamencoguitar3658]

1/sqrt(2pi) is also "incidentally" the normalization factor when you want to build an orthonormal basis of an infinite dimensional L2 space from the functions e^(ikx). This factor is what it takes to make the norm of each such element =1. And that's also how the area of the Normal gets calibrated to 1. So you could approach this from a number of angles... (Disclaimer, not a mathematician, but just sayin'.)

[@paradoxicallyexcellent5138]

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

[@hectorcelis678]

The same beauty in math and history: the best part is the story behind the facts.

[@dylanparker130]

How true!

[@histeward]

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

[@moralboundaries1]

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. ❤

[@user-vt6td9hp3g]

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.

[@nadaelnokaly4950]

yess please, who else need more stats videos? +1 here

[@JoshSmith-db2of]

Grant has said in the comments section of the other videos in this series that he anticipates making a stats series soon! These videos will be a part of it.

[@andrerenault]

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

[@patrickguest2762]

i was gonna say... also the light in that piece of art is something beautiful

[@tobiascornille]

I wonder if it's AI generated. Looks so to me, cause the hands are bad 😅

[@KernelLeak]

I thoroughly dislike the part where it says "aided by Midjourney"... yuck.

[@raptor4916]

@@KernelLeak Midjourney is just like any other artist, it learns by looking at 1000s of examples

[@Aredtyg]

​@@tobiascornille Well, I doubt there are too many paintings of statisticians explaining the Central Limit Theorem to their friends while at lunch...

[@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 :)

[@PianoPsych]

This video is breathtakingly marvelous! You elegantly answered several lingering questions that I thought I held separately and brought it all together in a visually stunning package. It was like a gift from heaven. You remind me of why I love mathematics so much. I'll be rehearsing this lesson in my mind for years to come. You've earned every penny of Patreon support I've given to date on this one video alone, and I have enjoyed so many others. Many, many thanks for sharing your extraordinary gifts with us.

[@coffeeguy.3438]

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

[@johnchessant3012]

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.

[@ahmedgabr8009]

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

[@v2ike6udik]

You can always add bombs and blisters :')

[@Joyexer]

It always blows me away not only how good your animations look, but also how well they underline the concept you are teaching!

[@mayurmatada6298]

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

[@Lavamar]

This was such a great video! Well paced, framed and explained. I only hope that the last part comes out soon. Excited!

[@steffenvraagidius1701]

Great video. I love the storytelling in this video. To me, one of the fun parts of working in the field of math, is the social aspect of sharing knowledge and the explaining and reasoning which gives me joy, and the pictures of the people talking in a café adds to that feeling. I have a freind, who everytime we meet, demands me to explain some math to him, so that he can get his mind blown. This video made me think of that.

[@VioletCatastrophe]

The problem at the end was a delight to solve honestly. Part 2 was a really fun extension of the initial trick in the video. Thanks for sharing it. The video itself was also equally great.

[@Rightsideup23]

YES YES YES! Thank you so much! When I took statistics, I always had this sense that I was missing something, because I never had the same intuition for it that I did for other areas of math. Something about this video just made everything click. Keep up the good work!

[@Me-0063]

Love the video! I really like how you explain such complex things so simply and in such short time!

[@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 .

[@mehtubbhai9709]

I LOVE THIS CHANNEL!! I have always wondered what pi was doing there. Thank you so much Grant for throwing back the covers. You truly have reawakened my wonder for Math

[@AriaHarmony]

It's teacher appreciation week, and I must name you one of the best teachers I have ever had, Mr Grant! Your calculus and linear algebra series opened doors for me, thank you so so much!!

[@Kynitt]

I think this is the best video you've ever made. I absolutely love the distinction you make between proofs that are beautiful (slick, elegant, clean) and proofs that provide intuition (clearly use the assumptions that they start with). I can't wait for the finale!

[@saltrocklamp199]

I keep finding in math that learning the history of how an idea was discovered serves to illuminate the idea in general, and makes the "modern" version of that idea so much more satisfying as well as easier to work with. Thank you as always.

[@AdamSmith-ne9jm]

I'm so incredibly happy you're doing the probability videos. Been waiting for this series since first discovering your channel 5 year ago! The linear algebra one boosted my gpa by probably a full point lol

[@artyom2801]

This has actually made me understand statistics better than my university when discussing Radiation measurements. It wasn't alone as I went out of my way to look up further information beyond the supplied reading material, but it definitely helped and motivated for me to do such, thanks for that.

[@benjaminclehmann]

The Herschel-Maxwell derivation is also a very nice justification for why Gaussian convolutions are separable (i.e. you can apply a gaussian blur to an image by blurring it only in the horizontal direction and then blurring that vertically).

[@kashyapbrahmandam3586]

i wish I could be a mathematician like you. The ability to intuitively explain these stuff is very uncommon and it really stems out of your admiration and curiosity towards the subject. Thank You!!

[@thelifeofahuman3666]

I have done engineering from a very lowly college but still, my engineering math teacher was so succinct in teaching exactly how you have taught with animation at that time I didn't care enough but now watching your video just made me realise there are good teachers in every corner but we just pass them and don't appreciate their hard work. Your channel always helps in learning new things and re-learning what is hidden inside our minds. Thank you so much for your contribution.

[@OwenMcKinley]

Thank you Grant I live in a time period where I can see this in my hand from the comfort of my couch. World class, exceptional I'm grateful!

[@HideD62]

AI인걸 못 느낄정도로 어마어마하게 자연스럽네요. 억양이 살짝 단조롭다는 느낌은 들지만, 대본을 읽는 사람들은 보통 이렇게 말하거든요.

[@user-le9xu2mf5g]

I really appreciate that you elaborate some trivial things, as, for example in 8:40. You spell out exactly, what taking an antiderivative value at point \inf means. Your manner of spelling things out in concise yet meaningful way is extremely helpful Keep up your incredible work, Grant!

[@Vitorruy1]

Im here for the Korean AI dub

[@smirnoff_the_best_vodka]

Man hold up, that's AI??

[@Ihate_rice]

​@@smirnoff_the_best_vodkayes, he said it in the post

[@honehan-yh3fj]

wow. amazing.. so realistic

[@unebonnevie]

I go to this channel whenever I want to feel elevated intellectually! Great stuff on math!

[@ColeCoug]

These gaussian integrals are all over the place in quantum field theory! My QFT homework also taugt me a fun relationship between the area of the unit sphere in D dimensions and the Gamma function. 2 pi^(D/2) / \Gamma (D/2) . Would love to hear your thoughts on renormalization sometime that stuff is WILD. The Zeta function even shows up sometimes. I've started to make my own visualizations for physics and I find them so helpful. Great video!

[@giacomocorvi7896]

Looks like the footnote at the end of the video can be generalized to non-integer dimensions

[@TheFireBrozTFB]

I love this, I made a project about volumes of spheres in higher dimensions in the past, and watching this video added a lot more clarity and understanding! Ty for the great content!

[@SalehFawaz]

From the moment I first learned about the fascinating concept of the Gaussian Integral, I had hoped that 3Blue1Brown would produce a video on the subject. And true to form, the video they created was truly exceptional. Grant's expertise and presentation style were truly captivating, and I am grateful for the opportunity to have learned from such a talented educator. Thanks Grant.

[@andrewpearce6943]

I'm guessing Grant is building up to a 23 minute video explaining a function that describes everything everywhere

[@lookupverazhou8599]

God.

[@BB-yi1oq]

all at once

[@shoam2103]

​@@BB-yi1oqalmost!

[@mdabusaqibalamansari9979]

Amazing video. Your explain things soo well. Maths seems so much fun with your videos

[@freudsbreakfast4060]

I absolutely love you and your videos. I'm a math and computer science major. Between the statistics of building neural networks and the math of fourier analysis that I've been studying in detail in school, these last three videos have given me such a synthesis and crystallization. Everything is connected, and math is the language that describes those patterns!

[@physicsguybrian]

Your presentation style and graphics are absolutely outstanding!!! A true pleasure to watch and learn from! Thank you!!!!

[@markkennedy9767]

Again, I'm blown away with how Grant is all about how someone could rediscover maths and is suspicious of tricks in doing maths. Without doubt, the best maths channel on KZbin. He just gets what learning maths is all about: how would someone rediscover for themselves why something is the way it is- to feel the maths in their bones, not just remember tricks. Just brilliant.

[@hak0bu]

Bro the Korean sounds so good. It is slightly low volume and muted, but overall very promising!

[@wpgg5632]

Really wonderful, both demonstration and your manner to explain is such a real beauty ! Thanks for that

[@manuelbeato13]

One of the best videos I have seen in quite some time. Loved the curious and pedagogical approach.

[@MadaxeMunkeee]

I’m glad you put “beyond integral tricks” in the title, otherwise I would have scrolled past lol. I didn’t know about Herschel’s derivation. This is a great video, thanks for making it!

[@12321dantheman]

Ohhh i just realised that that's why the maxwell boltzmann distribution is what it is. I like this a lot more than the proof we saw in uni, which iirc used quantum mechanics (which maxwell didn't know about). Very cool thanks

[@joekerr3638]

Love the user name

[@alperenerol1852]

This brings back so much memories. I remember the first time seeing this proof in my multidimensional calculus class. I was literally blown away when I connected all the dots. It’s so neat and trivial. I just realised how fun math was back in those days.

[@sphatak11]

This is incredible. Thank you for explaining with such amazing clarity. It will be hard to top this.

[@gabrielebartolini6679]

I don't know how this is possible, I was thinking about this thing right today, thank you 3blue1brown!

[@nothinginteresting1662]

If anyone is wondering how b^x can be written as e^(cx), it can be done because b^x = e^(ln(b^x)) = e^(x*ln(b)) = e^(x*c) = e^(cx), where c = ln(b) *Edit:* If c is negative, it implies that b lies between 0 and 1.

[@im-Anarchy]

what's the time stamp

[@nothinginteresting1662]

@@im-Anarchy Around 20:05 is when b^x is written as e^(cx)

[@im-Anarchy]

@@nothinginteresting1662 actually that was pretty easy but I am grateful If you are okay with solving my doubt which is. 14:44 how the hell f(x,y)=g(x)h(y)

[@nothinginteresting1662]

@@im-Anarchy g(x) and h(y) are distributions of x and y respectively. Then it is shown that g and h are equivalent because there is no difference after switching the axes. The reason for having g(x)h(y) is that x and y are independent of each other. If they were not, it would have reduced to a single variable function in either x or y.

[@im-Anarchy]

@@nothinginteresting1662 that's ok now let me ask a personal question are you a student or working profesinal , entrepreneur. are you successful and happy, because I want some career guidence

[@DroCaMk3]

Aside of the beautiful animations and very well-explained math, the art in this one was especially nice!

[@dsudikoff]

Love what you do! It's great to have a mathematician ask the "Huh?" and why are these connected questions . In my studies the sense of wonder and sublime beauty of the connectedness (as well as original joy of the how of discovery by the original mathematicians) was wrung out of the process. You ask the "meta" questions!

[@davidnoll9581]

A vaguely interesting little thing I realized a while back is you can write that as e^(-x)^(x) ... So there's a sense in which there's an exponential going in one direction, and then another going in the opposite direction. In this way, it feels to me to reflect 2 opposing exponential forces... Another way to look at it that is a little more circle-y is e^(i*pi)^(i*pi)... It works for matrices too (applying the linear transformation to rotate by 90 degrees twice yields something gaussian), and of course the infinite sums, but I guess that's all just kind of just recapitulating definitions. I spent way more time than I should have trying to figure out if there's anything deeper there and I couldn't find anything though

[@limit22]

you are part of the reason why the world will be smarter and achieve more. the impact you and your channel will have on people. I wish I had this growing up

[@firmkillernate]

Learning calculus has definitely influenced and changed my life. Thanks for the deep dive on one of my most inspirational problems in mathematics!

[@bramfran4326]

Hands-down perfect video. I am patiently waiting for the next one, it is going to be a revelation!

[@ttiff97]

5:32 One way I’ve solved the integral of e^(-x^2) is to use the Taylor expansion of e^x (this was during my university days). As mentioned, the anti derivative is non-elementary

[@Anvilshock]

"the question raised by the hypothetical statistician's friend" - You heard it, even 3b1b thinks statisticians have no real friends.

[@artforster4759]

Excellent and informative video! Thank you! I like your use of the Galton Board beads (which are independent of each other) as a dynamic way to illustrate adding together many different independent variables to produce a normal distribution via the IID CLT.

[@aarong2374]

Thank you for this video. I, so like many, wish I had this during my studies. I'm glad that this will be available for those new into this field.

[@IseOnCrack]

I love how you explained intuitively the Jacobi determinant for the polar transformation, you’re way of explaining math is amazing

[@dan-florinchereches4892]

Yeah that was fun. I remember me skipping class a lot in Uni and just cramming double integrals without going to class just trying to solve the exercises. One of them was triple integral for sphere volume and i had no good parametrisation with nice bounds in mind. And then i remembered a book about a tank crew positioning the turret by rotating the canon with 2 angles for azimuth and zenith. That was when i naively discovered polar transforms and even funnier I just naively represented x y and z as terms of R alfa and beta then pretended I did a variable change and just calculated dx dy and dz plugged in and somehow got the right result. Many years had passed until i found out the jacobian from algebra was actually not put there to torture us😂

[@miguelangelmartinezcasado8935]

I was this week reading a paper on how to create neural networks that outputted a measurement and a covariance. And that pi was really something weird in the loss function that I could not understand. Thanks for making math more bearable!

[@yuhanmao6512]

Hi Miguel, that sounds like a very interesting paper, would you mind sharing the name?

[@kostoffj]

@@yuhanmao6512 yes, please share

[@yecinemegdiche3202]

Always wondered why the normal distribution is the way it is. The explanation in this video is extremely satisfying. Thank you!

[@kayrasener5142]

Hi Grant! I'm an Industrial Engineering junior who's interested in pure math and statistics and altough I'm comfortable with their applications, I lack most of the technical knowledge and cenceptual understanding to fully absorb the essence of the notions' origins and proofs in these fields. I was surfing on the web about natural logarithms of complex numbers, which somehow lead me back to the central limit theorem and I pondered upon the very questions you examine in this video-essay. It's so refreshing to find such an expert source with a very fluid and graphic teaching method, explaining all the miniscule details that I was specifically curious of and many more like them about such sneak and elusive concepts. You're a real gem! Thank you for broadening the minds and enthusiasms of millions like me!

[@banaverhel]

I love how this channel is a gateway for amateur mathematics enjoyers (like me) to gain an intuition on a topic and be able to contextualize and explain these (admittedly rather complex) concepts. I, for example, am a high school student who loves learning about math but my teachers refuse to talk about anything other than basic high school math and give us hundreds of pages of excersises to mechanically compute. On the other hand, I cannot really understand the dense mathematical textbooks for university-level students as firstly, I do not even know the motivation or reasoning behind the proofs and derivations therein, but I also do not understand what the notation means in practice. However, your videos provide the bridge that lets an uninitiated, amateur mathematician understand and learn about these things. For example, after watching your calculus series like 3 times I finally understand the stuff in calculus textbooks and I could follow the proofs and the rigorous definitions there. Therefore, I think it’s no underestimation that you have changed the course of the most mathematics enthusiasts’ lives, including me, of any person ever. Thank you for your hard work!

[@danielagentile5355]

The quality is getting better and it was already perfect

[@haoming3430]

My gosh, this is just mind blowing. It just gives you a totally in-depth way to look into things.

[@MmmVomit]

Thank you for talking about the Herschel-Maxwell derivation of a Gaussian. On the last video I mentioned how use of the standard deviation formula always seemed very arbitrary and "handed down from on high" in all my science classes. I very much sympathize with the friend in the cafe questioning where pi came from. This video is starting to tie some of those unrelated threads together in my mind. Not well enough yet that I feel I could explain it to others, but well enough that I'm starting to feel satisfied with where it came from.

[@cycletofun5964]

Great video! Can you do a video on Bessel's correction and "degrees of freedom" in statistics in general?

[@Ricocossa1]

1:30 I remember when I saw for the first time that you could derive the formula for higher dimensional spheres in that way. It was in Peskin&Schroeder's book on QFT. They have this two-liner in the middle of a computation where they show that result, and I remember thinking why I'd never thought of that before.

[@Ryuuuuuk]

It also pops up in introduction to statistical mechanics in the derivation of Boltzmann entropy of an ideal gas. In the microcanonical ensemble we coun't all the states that have the same energy or put otherwise are in the same shell of a multidimensional sphere. I like that this video exemplified that the Gaussian stems from radially symmetric and uncorrelated, which is just the ideal gas.

[@andrewharrison8436]

24:56 that's a few seconds of results that I never knew I needed - screen printed for later. Yes, I knew all this from err... 50 years ago but I never saw how it fitted together, so satisfying - thanks.

[@GeorgeZoto]

Amazing explanation and funny narrative 😃 Thank you for creating such beautiful content, keep it up 😃

[@martincook8770]

@3blue1brown As you’ve already talked about normal distribution, can you also talk about estimation theory (ex. maximum likelihood estimation, bayesian estimation) and hypothesis testing such as likelihood ratio test or wald test?

[@spinyslasher6586]

This is weirdly relevant for me, cause I'm doing a course on Statistical Mechanics and we're dealing with this exact problem of deriving the gaussian distribution. Thanks a lot 3b1b!

[@syuhaichen3273]

I can finally feel that the definition of normal distribution is worked out by human beings instead of coming out of thin air, what a great explanation.

[@brunoheblingvieira5141]

Very nice video, just one point (which I guess most know, but still) about Maxwell's equations: while Maxwell derived the laws, it was Heaviside who later wrote the equations. Maxwell didn't have Vector Calculus, so he had tens of equations for the electromagnetic phenomena. Heaviside was able to summarize them all into the four we know with his (independent) discovery of Vector Calculus principles.

[@sleepycritical6950]

The artwork is really good. I hope I can one day draw like that...and also the math is beautiful too

[@l3gacyb3ta21]

Sorry to say that it’s, at least in part, ai generated

[@sanderson84060]

@@l3gacyb3ta21 Actually, it isn't. Purely human.

[@cerocero2817]

.​ @sanderson84060 What leads you to that conclsion? The description claims it was made with aid from Midjourney

[@marco_burderi]

Grant, thanks a lot for your work. You're really just one of a kind.

[@vmiguel1988]

I watch those videos mostly because or the beauty of your explanations, I understand a bit of them but the holes in basic mathematics concepts I have after growing in a country with poor basic education will never let me fully appreciate the full extent of your message

[@whiterook6]

I couldn't figure out how f2 turned into g * h at 14:58. Then I remembered this is probability, and that g and h are the probability of two independent events, so of course it turns into a product.

[@smotala11]

Haha I paused at that point too before realising.

[@brianprzezdziecki]

These videos are timeless and will be used for generations to come

[@lookmath4582]

It is one of the most elegant videos of you that I have ever seen . It is packed with information and beautiful techniques . It definitly refuels my math stamina ❣ .

[@haoding1812]

3b1b's video is the best video I have ever enjoyed while having lunch alone. However, I have lunch alone more often than you post videos. Either you need to make videos more often or I need to make some new friends.

[@nazir221]

Listened to in Korean audio, can't understand it, but sounds sweet. Good initiative 3b1b

[@christopherryzowicz6684]

Grant, you should do a video about the new proof that 2 high schoolers came up with for the Pythagorean Theorem! It’s in the news, easy to find and a great discover for to young mathematicians.

[@abhishuoza9992]

I didn’t even know it was possible to have climaxes in mathematical derivations, yet these 3 videos make me feel just like watching a tv series. I want to know how it ends! Btw I am loving this series. The Gaussian distribution has intrigued me since I first heard about it, I always wanted to know why aggregation of random processes end up making a bell curve. Keep doing the great work!

[@jacksonstarky8288]

At the start of the video I was suspecting that Euler's formula might be involved... but this was far more interesting... and the illustration of the 3D bell curve was an absolute eureka moment; of course there has to be a circle there. Looking forward to the next video tying it all together.

[@ridazouga4144]

Man it's so unfair that you didn't get a mathematical prize and recognition for creating this historical channel

[@ddognine]

23:25 "The Central Limit Theorem which is all about adding together many different independent variables." One interesting artifact of the CLT is the requirement of adding the variables. If, for instance, you multiply the variables together, you get a lognormal distribution instead. You can see this for yourself by calculating the expected probabilities of two dice whose result is multiplied. Therefore, the CLT is most applicable for data that varies by only a single order of magnitude. If you add two dice, the largest result is only 6X the smallest result which is inside a single order of magnitude, but if you multiply the dice, the largest result is 36X the smallest result. So one way to determining whether your data really follows a normal distribution is to calculate the "order of magnitude" present in the data. Because the normal distribution decays so quickly it can't model widely varying data very well and you will likely need a fat-tailed distribution. This has bitten many "experts" when the probability of extreme events is much larger than what is implied by a normal distribution.

[@Miguel_Noether]

Link for examples please

[@AuratticStride]

I'm really enjoying this series of videos, and I love the way you explain things down to the very last detail. Also, I hate to be the umm actually guy, but at 21:40, the equations you show (the ones we us in the modrn era) are the ones derived by Oliver heaviside, after applying vector calculus to maxwell original 20 equations

[@innercalmmeditation]

The video clearly explains the fascinating phenomenon of how the value of π appears in the normal distribution. It's fascinating to see how this fundamental constant relates to the bell curve, and it's a great example of how mathematics often reveals surprising links between seemingly unrelated concepts. The explanation is both accessible and informative, making it an excellent resource for anyone interested in understanding the beauty of mathematics. Overall, a fantastic video that sheds light on an intriguing aspect of probability theory.