The follow-up to explain how this fits into the central limit theorem: ruclips.net/video/d_qvLDhkg00/видео.html That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: ruclips.net/video/IaSGqQa5O-M/видео.html
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
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
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.
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
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
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 👏
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.
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 .
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.
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
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.
@@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)
@@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.
@@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
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!
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
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!
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.
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.
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!
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.
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.
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.
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.
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!
My compliments to the artist and the inspiration to use their beautiful watercolors to merge abstract mathematics and humanity. Beautiful artwork and wonderfully complimentary to the elegant graphics and insights.
I don't know if you have read the description, seems like it was mostly Ai generated, which isn't a bad thing but seems things are changing bit too fast
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
Ohh yes! This would be so good. I'm getting ready for college next year, and there is definitely a lack of resources in that area. Really would like an approach to Linear Algebra and Multivariable Calculus that really is always focusing on how these new ideas are analogous to our previous understanding, not just some absurd new construction.
Didn't he do one way back on Khan Academy? Don't get me wrong, I'd love to see a new "Essence of Multivariable Calculus" as a sequel to his series on Calculus.
Oh my God yes...That's like the one huge foundational topic he has yet to make a series on. An area like that is something that would really benefit from his brilliant visuals.
I did make around 100 videos on the topic when I was at Khan Academy. I'm not opposed to doing some sort of "essence of" series on the topic in the 3blue1brown style, but it's not as high on the priority queue as some other things.
Another condition like the Herschel-Maxwell derivation that basically "uniquely determines" the normal distribution is that the distribution of a linear combination of normal random variables is itself the same linear combination of their distributions. That is, if we have two distributions X and Y, and a random variable x and y from each, and a positive constant c, then x + cy has distribution X + cY which is itself a normal distribution. I noticed this when designing a ranking system similar to ELO but with a normal distribution instead of a logistic distribution. In that context, it means that the posterior distribution for a normal distribution conditioned on the output being larger/smaller than some other normal distribution is again a normal distribution. But the "closed under linear combinations" property is useful in a lot of other contexts, for example, because it is exponential it is equivalent to Hershel's second property. This also gives great insight into why the normal distribution shows up so frequently. In the central limit theorem, we basically see that a sum of random variables taken from the same distribution will approach a normal distribution. But if we think of the sum not as a sum of the original random variables, but as a sum of a few sub-sums of the original random variables, then by the central limit theorem, each of the sub-sums will have a normal distribution, so the sum of the sub sums will have a normal distribution. This looks like using the central limit theorem to prove itself, and it is, but that's because it isn't meant as a proof but rather to lend intuition. What this means is that the more random variables you add, the less normal the originals have to be for the sum to be close to normal. And the fact that we could split up the sum into sub sums shows that even when the problem looks one dimensional, it's really n-dimensional, since it is the sum of n original variables.
Eugene Wigner's philosophical approach was very profound. The "unreasonable effectiveness of mathematics..." is an amazing paper to say the least ! Whenever I discuss math and the universe, the conversation always tends towards this result
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.
I must admit, I struggled a bit with this one. I could've done with an explanation of the integral at 8:10. I can accept that 2r•e^(-r^2) is the integral of -e^(-r^2) because it follows roughly the right visual pattern I remember from calculus over a decade ago, but a few seconds of the equation parts moving around to show the derivation would be much more satisfying than just accepting it. It also would've been nice to see a quick explanation of why b^x and e^(cx) are equivalent in this case at 20:00. Again, I can trust that it is true, with some vague recollection of exponents behaving like that, but seeing why and how would help avoid that diversion of cognitive load away from the topic at hand, and lead to a much more satisfying resolution. With all that being said, I did mostly understand the overall point being made, and I'm eager to see this tie back into convolutions, especially if you can tie it to a real-world application like image or audio processing.
Chain rule for -e^(-r^2) Derivative of the outer function multiplied by derivative of the inner function Outer function is of the form e^x, whose derivative is the same (e^x). Thus, our outer function’s derivative is -e^(-r^2) The inner function is of the form x^n. When calculating the derivative of x^n, you obtain nx^(n-1). In the case of -r^2, you’d obtain -2r^1, which is simply -2r. Multiplying our outer and inner derivatives gets us -e^(-r^2) * -2r = 2r * e^(-r^2), which was our integral. Thus, -e^(-r^2) is the anti derivative of 2r * e^(-r^2) Hope that could at least clarify the first part! Don’t know enough about e^cx
12:00 the most convoluted proof I've encountered. It really feels like a trick. Like proving something by not really proving, but proving in another way that depends on the previous proof. Like a mutually recursive proof
I'm a chemist by training and the integral of e^(-x^2) was the only time in the two semesters of mathematics when I was thinking "Wow, that's cool" during a lecture. We used a bit different method of transforming the double integral (in x and y) into polar coordinates (in r and theta) using Jacobian. But the result was the same, of course.
As someone who already knew multivariable calculus, integrals in 2 dimensions and polar integrals, once you showed the step up a dimension I immediately knew what was going to happen next - but the explanation of how we get the formulas for the volume was the clearest I've ever seen and is a better introduction to multiple integrals than my multivariable calculus course.
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!
In Second part you explained the reason for pi to come up in a Gaussian distribution was due to circular symmetry, but take a example of what Maxwell was doing studying vel of gas mol. in a vol say that vol is a cube then there won't be circular symmetry, right? So there shouldn't be pi in that distribution also same goes for population distribution on a map is not circular symmetric so how does pi come up there ?
Ah, yes, I spot the Witch of Agnesi. I remember in university stats class where the prof pointed out that sigmoid functions (like arctan) as cumulative distributions lead to bell curves. But just because it looks like one doesn't mean that it behaves well like a normal curve. The arctan gives you the witch curve, and if you generate values in that distribution, they'll look normal for a little while, and then suddenly give a value that seems many sigma out. Except not really, because that distribution is pathological, with no defined mean or variance.
As i like to say, Mathematics is the science of certainty and Statistics is the science of uncertainty! Perfect nighttiime video✨!! Would like to see more on theoretical inferential statistics from u(esp. sufficiency criteria)!
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.
I convinced myself(intuitively) without rigorous elementary proof why pi is there in formula is because of area/volume we are able to calculate in multivariate Gaussian. One can look this through by looking at the projection plot of multivariate gaussian e.g. p(x) and p(y) together. Otherwise re-imagining gaussian integral in multidimensional context requires lot of hard work which as usual Grant has undertaken this challenge to take us through this beautiful elementary journey.
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.
There are so many things I don't know, it feels like mathematicians are on a whole other level and I am very glad to have such a good quality content creator who shares his love with such a passion. Here an idea that sprouted from watching this video, it is a bit off-topic but I would still like to share it with whoever reads this. In a way the world has always been about building and deconstructing things, be it the trees that grow and burn away to give soil for the next generation. Science, where we build a model until it fails, from which we learn and build a more refined model. Or more closely related to us, civilisation that rise and fall as time passes. Things are bound to "fail", or rather change. It is a constant that we have to take into account when building today. The ability to deconstruct is something that is slowly being integrated into the product design. We are gearing towards construction and destruction as efficient and easy to do. Symmetrical, like a sine function, two side of the same coin.
I just simply started learning normal distribution but was NOT satisfied with the explanation in the book for Probability density function (PDF) and Cumulative density function, I just started digging and asking why's on how the formula of PDF was deduced. And now I am watching this video. Mathematics is a dark and GORGEOUS !!!
I knew it hasn't ended yet, but thanks for making these series. This particular topic for me connects computer science (convolution), probability theory (central limit theorem), and gravitational physics (tensors and geometry). From the first video about convolution, I knew that it will lead to this gaussian area integration, so I was looking forward to how you will break it down to a more down-to-earth derivation. In the past, it all clicking together for me when I realized that the independence of dimensional geometrical axis means that the probability of events in those axis were independent as well. This naturally causes pi to be involved, since circle or n-sphere is the maximum amount of boundaries with these "independent" axis as constraints. Your convolution video made realized that any specific distribution in quantum level might be discrete and skewed. But if the operator can be infinitely added to a very far dimensionality, it should averaged as a sphere due to central limit theorem. So whatever physical effects observed in a sufficiently far distance will have interaction shape like a gaussian probability. In this point of view, the probability of the small scale interaction/events gave birth to the large scale geometry.
It always feels like, to me at least, the people who talk about maths' "unreasonable effectivness" kind of subtly ignore things like the n-body problem, the Abel impossibility theorum, Godel's paradoxes, etc. Is maths actually unreasonably effective, or are we just excluding the ways in which it's ineffective?
The unsolvability of the quintic and Gödel’s incompleteness theorems are both quite irrelevant outside of pure mathematics. Considering the ridiculous amount of problems math has been able to solve, it is clearly effective, some might say unreasonably effective. The fact that a few fairly irrelevant parts don’t work out perfectly does not change this fact. Really the only major problem is chaos, which we can better understand using, you guessed it, mathematics.
@@arnouth5260 Literally the _entire point_ of the "unreasonable effectivness" of maths is that elements of pure mathmatics are useful outside of maths. The fact that these topics are "quite irrelevent outside of pure mathematics" _means_ that they are examples of maths being ineffective. Maths is fundamentally the study of how abstract systems behave and interact. If you take any natural phenomemon, _and abstract it,_ you can apply mathematical reasoning to it. That's not unreasonable effectiveness, that's the thing doing the thing it was designed to do. Besides, if you cared to take a step outside of STEM and looked into, say, sociology or economics, you'd find maths is often not only ineffective but actually dangerous. The damage that Chicago school economists have done with their mathematical models is probably, ironically, incalculable.
@@Eden_Laika unreasonable effectiveness means math is far more useful than you would expect. The fact there are some things which aren’t effective doesn’t change this. You wouldn’t call language useless just because a sentence like “Green dreams sleep in purple beds.” Is meaningless would you? Neither would you claim it’s useless because propaganda exists. No one is saying all of mathematics is useful in every single circumstance, hell the “unreasonable effectiveness” applies to the natural sciences, that’s literally what the original article was (mainly) about. The fact you don’t see how that effectiveness is surprising just goes to show how little mathematics or science you actually understand.
@@Bagginsess Economics is a social science that _thinks_ it's a physical science. Maths has its place in economics, as statistics, but the proliferation of mathematical economic models had been deletrous to the good of the world; see Chicago School, Reaganomics, Thatcherism, etc, etc.
At uni a professor gave a quite simple, perhaps even trivially so, explanation of why it's precisely e^-x^2 and not something else: Something in nature tends to be the sum of random parameters, and the sum of random parameter tends to follow that curve. (The more random parameters are summed to the total, the more closely the curve will approach that function.)
I love math because it is useful and accurate, and my friend loves it because a single spark of genius, usually seemingly unrelated and random, could make things a lot easier. Needless to say, the fact you answered these questions in the order that would be explained in your example conversation amazed us both
What a shame you've resorted to ai art. good math tho in case this comment somehow actually gets seen and people cry about me dissing ai art: the issue isnt whether its "real" art or not, you can't define what "real" art is in a way that encompasses all art and excludes all not-art. the issue is how it gets its data, usually by stealing artists' work without their permission or knowledge.
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 ❣ .
I have been teaching this derivation of the volume of a D-dimensional sphere for years, it is needed in statistical mechanics, when deriving the properties of the perfect gas within the microcanonical ensemble. As a side product, one can show that the ratio between the volume of a shell of ARBITRARILY small (but FIXED) thickness and the volume of the ENTIRE sphere tends to 1 as the dimension D tends to infinity (the volume of an infinite dimensional orange is concentrated in its peel)
It is such a delight to watch these videos... Really love how you bring the intuition of Pi into the minds of the uninitiated... BTW, 'Pi Day' is just around the corner. Our condo management is organizing a 'Pie party' (of course at 3:14 PM) - how sad!! (this is somewhat like celebrating Gandhi's birthday with a candy party..:)) Would be great if you can suggest some real activities for the day that respects this amazing number...
For the animation at 21:10 - 21:30, why do the corners* start to shoot up before c becomes positive? For all c0 the corners* should shoot up to infinity…. *I know the graph extends out infinitely far from the center - by “corners” I mean the corners of the section of the graph that are visible in the animation.
Dear 3Blue1Brown, I'm a fan of your contents but I rarely comment. My comment here is to bring your attention to a potential mistake in this video production of yours. I'm referring to the part containing AI-generated pictures. I immediately spotted them as AI-generated, because they are full of artifacts: distorted and unnatural faces for the two ladies; ugly hands; cadaver-pale colors in Maxwell's face; hard edges everywhere; VERY ugly pixelization and JPEG-like artifacts all around. This feels cheap. Very cheap. And this uncomfortable feeling is inevitably impacting the viewer's perception of your content. And, as a consequence, the sense of worthiness attributed to your work. There is a reason why good artwork is costly and time-consuming. And, if you really appreciate the value that artworks can bring to your contents, that's exactly why you shall consider hiring a proper artist. Don't fall in the cheapness trap. Safeguard your contents. [Disclaimer: my PhD thesis is about ML architectures, so I'm well aware of AI limitations.]
Hi Federico, I hear your concern. For what it's worth, the goal certainly wasn't to be cheap; I did hire an artist to take care of those scenes, and we decided to experiment with using Midjourney in the spirit of better engaging with the full scope of tools out there, though the artist still had a very active hand in deciding on the overall concept and refining what Midjourney offered as a starting point. My opinion on AI art is still not fully formed, and I appreciate your feedback. I'm curious, do you have an objection to the premise of using these tools at all, or if the artifacts were sufficiently addressed by the artist, would that be enough for it to feel alright?
@@3blue1brown My opinion on AI art is still not fully formed either... We all know that those visual artifacts will surely progressively diminish with time. For now, I think they are just ugly and distracting (and feeling "cheap and quick", as I told you in the previous comment). Nonetheless, I believe that untill there is a serious law framework regarding the copyright issue, many people out there will see the use of those tools as a sort of "theft". And there are many artists in the science community (your viewers)... I'm not an artist, so I cannot speak for them, but I can surely see their point. Regarding me personally, I really like when things are well made: with intelligence, vision and taste. Being it a gorgeous oil painting or a insightful mathematical video, it's like having the possibility to breath fully and profoundly. I think the aim should be to amaze and delight, chasing the best.
Why the heck are you so smart. I don't get it. How do you figure out the most intuitive and insightful ways to explain things? You're incredible Grant. I'm baffled. Not by the video (your explanation makes complete sense), but by you in general.
Your video s are so beautiful and and the visual concepts are so illuminating , that almost every single time i forget to give a like .... and i think this is true for many viewers of your creation ...
I had a VISCERAL reaction to the announcement that this would have another video. I was sitting there getting by pumped up for the answer about the two dimensions thing and then when he said I wasn’t going to get it, I reacted like a four year old who was told he couldn’t open his Christmas presents. Literal tantrum. 10/10 Would Recommend.
We think about Pi as the ratio of circumference to diameter because that is the simplest example and we learn it the earliest. In reality Pi has to do with continuous change - a circle is just a line that is continuously turning in on itself at a constant rate.
Stern-Brocot tree has the nice property that when we transform the fraction a/b to "simplicity" 1/ab, then all simplicities of each new row add up to one. Denominators of the simplicities have natural interpretation as parallelograms aka outer products. When we further interprete Stern-Brocot rows as winding numbers with circular curvature increasing from row to row, and keeping in mind that the density of fractions drops dyadically between whole number intervals, there seems to be much less need for probability theory approach...
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
The follow-up to explain how this fits into the central limit theorem: ruclips.net/video/d_qvLDhkg00/видео.html
That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: ruclips.net/video/IaSGqQa5O-M/видео.html
Still, why is the 1/2 in the exponent?
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
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
"Who ordered another dimension" 😂 classic mathematician path to solving a problem
Only PhD mathematicians have enough math money to order the 11 dimensions needed for string theory
@@InTrancedState We only get 3 and these elites get 11
Unfair
“Sir, this is a Wendy’s”
I'll have 4 dimensions with extra dip.
String theorists af
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.
raise this man to the top of the comments.
@@oszkarvarnagy7896 fax
Same
god damn ma man be talking like nepolean
@@jaw0449I’ll
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
Three parts blue 🔵, one part brown 🟤
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
Does your bully call you blue butt too?
what are the other 9
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 👏
I should inform you that this is AI lol
@@lifinaleWOW I think he had no idea when he wrote that
@@lifinaleYes… they said his AI voice…
@@Karlswebb they edited the post…
The Korean version sounds very natural. The pipeline works incredibly well!
Finally, the much awaited 3b1b statistics series is on a roll!
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.
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 .
I love the pieces of art that aren’t normally part of the “aesthetic” of 3b1b but somehow still fit right in.
i was gonna say... also the light in that piece of art is something beautiful
I wonder if it's AI generated. Looks so to me, cause the hands are bad 😅
I thoroughly dislike the part where it says "aided by Midjourney"... yuck.
@@KernelLeak Midjourney is just like any other artist, it learns by looking at 1000s of examples
@@tobiascornille Well, I doubt there are too many paintings of statisticians explaining the Central Limit Theorem to their friends while at lunch...
I'm guessing Grant is building up to a 23 minute video explaining a function that describes everything everywhere
God.
all at once
@@BB-yi1oqalmost!
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
Just wanna Say, Thanks for doing whatever you are doing. Never stop 3B1B
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.
It always blows me away not only how good your animations look, but also how well they underline the concept you are teaching!
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
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.
what's the time stamp
@@im-Anarchy Around 20:05 is when b^x is written as e^(cx)
@@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)
@@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.
@@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
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!
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
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!
This was such a great video! Well paced, framed and explained. I only hope that the last part comes out soon. Excited!
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.
I go to this channel whenever I want to feel elevated intellectually! Great stuff on math!
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.
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!
Hi Miguel, that sounds like a very interesting paper, would you mind sharing the name?
@@yuhanmao6512 yes, please share
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.
Haha I paused at that point too before realising.
I don't know how this is possible, I was thinking about this thing right today, thank you 3blue1brown!
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.
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.
Amazing video. Your explain things soo well. Maths seems so much fun with your videos
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.
Love the video! I really like how you explain such complex things so simply and in such short time!
Your presentation style and graphics are absolutely outstanding!!! A true pleasure to watch and learn from! Thank you!!!!
Man it's so unfair that you didn't get a mathematical prize and recognition for creating this historical channel
Grant, I want to thank you wholeheartly for this. Your previous video and this one made it click for me.
Listened to in Korean audio, can't understand it, but sounds sweet.
Good initiative 3b1b
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!
The truly eye-pleasing art style at the beginning fits nicely with the glibness of the rotational symmetry of the function under consideration.
Hands-down perfect video. I am patiently waiting for the next one, it is going to be a revelation!
My compliments to the artist and the inspiration to use their beautiful watercolors to merge abstract mathematics and humanity. Beautiful artwork and wonderfully complimentary to the elegant graphics and insights.
I don't know if you have read the description, seems like it was mostly Ai generated, which isn't a bad thing but seems things are changing bit too fast
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
Wonderful video, as always! Would you consider doing a series on Multivariable Calculus?
Ohh yes! This would be so good. I'm getting ready for college next year, and there is definitely a lack of resources in that area. Really would like an approach to Linear Algebra and Multivariable Calculus that really is always focusing on how these new ideas are analogous to our previous understanding, not just some absurd new construction.
I second that motion!
Didn't he do one way back on Khan Academy? Don't get me wrong, I'd love to see a new "Essence of Multivariable Calculus" as a sequel to his series on Calculus.
Oh my God yes...That's like the one huge foundational topic he has yet to make a series on. An area like that is something that would really benefit from his brilliant visuals.
I did make around 100 videos on the topic when I was at Khan Academy. I'm not opposed to doing some sort of "essence of" series on the topic in the 3blue1brown style, but it's not as high on the priority queue as some other things.
The best explanation I've seen anywhere. Thank you for making this video.
this video is amazing. and i should praise IT, but my god is the artwork for the statistitian and her friend STUNNING
it is too beautyfull i can't
Don't hang out near the top of the bell curve. Everyone there is mean.
Aside of the beautiful animations and very well-explained math, the art in this one was especially nice!
Could you put this all together in one playlist like your calculus series
Another condition like the Herschel-Maxwell derivation that basically "uniquely determines" the normal distribution is that the distribution of a linear combination of normal random variables is itself the same linear combination of their distributions. That is, if we have two distributions X and Y, and a random variable x and y from each, and a positive constant c, then x + cy has distribution X + cY which is itself a normal distribution.
I noticed this when designing a ranking system similar to ELO but with a normal distribution instead of a logistic distribution. In that context, it means that the posterior distribution for a normal distribution conditioned on the output being larger/smaller than some other normal distribution is again a normal distribution.
But the "closed under linear combinations" property is useful in a lot of other contexts, for example, because it is exponential it is equivalent to Hershel's second property.
This also gives great insight into why the normal distribution shows up so frequently.
In the central limit theorem, we basically see that a sum of random variables taken from the same distribution will approach a normal distribution. But if we think of the sum not as a sum of the original random variables, but as a sum of a few sub-sums of the original random variables, then by the central limit theorem, each of the sub-sums will have a normal distribution, so the sum of the sub sums will have a normal distribution.
This looks like using the central limit theorem to prove itself, and it is, but that's because it isn't meant as a proof but rather to lend intuition. What this means is that the more random variables you add, the less normal the originals have to be for the sum to be close to normal. And the fact that we could split up the sum into sub sums shows that even when the problem looks one dimensional, it's really n-dimensional, since it is the sum of n original variables.
This video and the way you connect everything...amazing! Every video I watch from you makes me wonder why didn't I go for math instead of engineering.
Eugene Wigner's philosophical approach was very profound. The "unreasonable effectiveness of mathematics..." is an amazing paper to say the least ! Whenever I discuss math and the universe, the conversation always tends towards this result
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.
I must admit, I struggled a bit with this one. I could've done with an explanation of the integral at 8:10. I can accept that 2r•e^(-r^2) is the integral of -e^(-r^2) because it follows roughly the right visual pattern I remember from calculus over a decade ago, but a few seconds of the equation parts moving around to show the derivation would be much more satisfying than just accepting it. It also would've been nice to see a quick explanation of why b^x and e^(cx) are equivalent in this case at 20:00. Again, I can trust that it is true, with some vague recollection of exponents behaving like that, but seeing why and how would help avoid that diversion of cognitive load away from the topic at hand, and lead to a much more satisfying resolution.
With all that being said, I did mostly understand the overall point being made, and I'm eager to see this tie back into convolutions, especially if you can tie it to a real-world application like image or audio processing.
Chain rule for -e^(-r^2) Derivative of the outer function multiplied by derivative of the inner function
Outer function is of the form e^x, whose derivative is the same (e^x). Thus, our outer function’s derivative is -e^(-r^2)
The inner function is of the form x^n. When calculating the derivative of x^n, you obtain nx^(n-1). In the case of -r^2, you’d obtain -2r^1, which is simply -2r.
Multiplying our outer and inner derivatives gets us -e^(-r^2) * -2r = 2r * e^(-r^2), which was our integral. Thus, -e^(-r^2) is the anti derivative of 2r * e^(-r^2)
Hope that could at least clarify the first part! Don’t know enough about e^cx
Well done Ju Ha-jin
12:00 the most convoluted proof I've encountered. It really feels like a trick. Like proving something by not really proving, but proving in another way that depends on the previous proof. Like a mutually recursive proof
You just add a dimension and leverage some symmetry you didnt realise was present.
I'm a chemist by training and the integral of e^(-x^2) was the only time in the two semesters of mathematics when I was thinking "Wow, that's cool" during a lecture. We used a bit different method of transforming the double integral (in x and y) into polar coordinates (in r and theta) using Jacobian. But the result was the same, of course.
As someone who already knew multivariable calculus, integrals in 2 dimensions and polar integrals, once you showed the step up a dimension I immediately knew what was going to happen next - but the explanation of how we get the formulas for the volume was the clearest I've ever seen and is a better introduction to multiple integrals than my multivariable calculus course.
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!
In Second part you explained the reason for pi to come up in a Gaussian distribution was due to circular symmetry, but take a example of what Maxwell was doing studying vel of gas mol. in a vol say that vol is a cube then there won't be circular symmetry, right? So there shouldn't be pi in that distribution also same goes for population distribution on a map is not circular symmetric so how does pi come up there ?
Really wonderful, both demonstration and your manner to explain is such a real beauty !
Thanks for that
Ah, yes, I spot the Witch of Agnesi. I remember in university stats class where the prof pointed out that sigmoid functions (like arctan) as cumulative distributions lead to bell curves. But just because it looks like one doesn't mean that it behaves well like a normal curve. The arctan gives you the witch curve, and if you generate values in that distribution, they'll look normal for a little while, and then suddenly give a value that seems many sigma out. Except not really, because that distribution is pathological, with no defined mean or variance.
Without even having watched a full 30 seconds, I gotta say, those are some beautiful paintings!
As i like to say, Mathematics is the science of certainty and Statistics is the science of uncertainty! Perfect nighttiime video✨!! Would like to see more on theoretical inferential statistics from u(esp. sufficiency criteria)!
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.
Thanks for the great video. it reminds me of the Monte Carlo method which finds the Pi in the random distributions
I will continue my journey as a Phd in Number Theory because of this Channel, wish me luck and thank you so much for your videos
I convinced myself(intuitively) without rigorous elementary proof why pi is there in formula is because of area/volume we are able to calculate in multivariate Gaussian. One can look this through by looking at the projection plot of multivariate gaussian e.g. p(x) and p(y) together. Otherwise re-imagining gaussian integral in multidimensional context requires lot of hard work which as usual Grant has undertaken this challenge to take us through this beautiful elementary journey.
Amazing! I have been wondering about this for so many years….
"Why π is in normal distribution?" A question I never really ask, but but should !
your Korean language is so good
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.
There are so many things I don't know, it feels like mathematicians are on a whole other level and I am very glad to have such a good quality content creator who shares his love with such a passion.
Here an idea that sprouted from watching this video, it is a bit off-topic but I would still like to share it with whoever reads this.
In a way the world has always been about building and deconstructing things, be it the trees that grow and burn away to give soil for the next generation. Science, where we build a model until it fails, from which we learn and build a more refined model. Or more closely related to us, civilisation that rise and fall as time passes.
Things are bound to "fail", or rather change. It is a constant that we have to take into account when building today. The ability to deconstruct is something that is slowly being integrated into the product design. We are gearing towards construction and destruction as efficient and easy to do. Symmetrical, like a sine function, two side of the same coin.
Thanks for making today just a little bit brighter! So much insight I missed out on in the first half century of my life ;)
Learning calculus has definitely influenced and changed my life. Thanks for the deep dive on one of my most inspirational problems in mathematics!
on a not so much related note - i love the artwork you used in this video :)
I just simply started learning normal distribution but was NOT satisfied with the explanation in the book for Probability density function (PDF) and Cumulative density function, I just started digging and asking why's on how the formula of PDF was deduced. And now I am watching this video. Mathematics is a dark and GORGEOUS !!!
I knew it hasn't ended yet, but thanks for making these series.
This particular topic for me connects computer science (convolution), probability theory (central limit theorem), and gravitational physics (tensors and geometry).
From the first video about convolution, I knew that it will lead to this gaussian area integration, so I was looking forward to how you will break it down to a more down-to-earth derivation.
In the past, it all clicking together for me when I realized that the independence of dimensional geometrical axis means that the probability of events in those axis were independent as well.
This naturally causes pi to be involved, since circle or n-sphere is the maximum amount of boundaries with these "independent" axis as constraints.
Your convolution video made realized that any specific distribution in quantum level might be discrete and skewed. But if the operator can be infinitely added to a very far dimensionality, it should averaged as a sphere due to central limit theorem. So whatever physical effects observed in a sufficiently far distance will have interaction shape like a gaussian probability. In this point of view, the probability of the small scale interaction/events gave birth to the large scale geometry.
It always feels like, to me at least, the people who talk about maths' "unreasonable effectivness" kind of subtly ignore things like the n-body problem, the Abel impossibility theorum, Godel's paradoxes, etc. Is maths actually unreasonably effective, or are we just excluding the ways in which it's ineffective?
The unsolvability of the quintic and Gödel’s incompleteness theorems are both quite irrelevant outside of pure mathematics. Considering the ridiculous amount of problems math has been able to solve, it is clearly effective, some might say unreasonably effective. The fact that a few fairly irrelevant parts don’t work out perfectly does not change this fact.
Really the only major problem is chaos, which we can better understand using, you guessed it, mathematics.
@@arnouth5260 Literally the _entire point_ of the "unreasonable effectivness" of maths is that elements of pure mathmatics are useful outside of maths. The fact that these topics are "quite irrelevent outside of pure mathematics" _means_ that they are examples of maths being ineffective.
Maths is fundamentally the study of how abstract systems behave and interact. If you take any natural phenomemon, _and abstract it,_ you can apply mathematical reasoning to it. That's not unreasonable effectiveness, that's the thing doing the thing it was designed to do. Besides, if you cared to take a step outside of STEM and looked into, say, sociology or economics, you'd find maths is often not only ineffective but actually dangerous. The damage that Chicago school economists have done with their mathematical models is probably, ironically, incalculable.
>maths is ineffective in economics
what?
@@Eden_Laika unreasonable effectiveness means math is far more useful than you would expect. The fact there are some things which aren’t effective doesn’t change this. You wouldn’t call language useless just because a sentence like “Green dreams sleep in purple beds.” Is meaningless would you? Neither would you claim it’s useless because propaganda exists.
No one is saying all of mathematics is useful in every single circumstance, hell the “unreasonable effectiveness” applies to the natural sciences, that’s literally what the original article was (mainly) about. The fact you don’t see how that effectiveness is surprising just goes to show how little mathematics or science you actually understand.
@@Bagginsess Economics is a social science that _thinks_ it's a physical science. Maths has its place in economics, as statistics, but the proliferation of mathematical economic models had been deletrous to the good of the world; see Chicago School, Reaganomics, Thatcherism, etc, etc.
If my highschool friend asked me what π is I would order a new dimension to lock them away in forever
At uni a professor gave a quite simple, perhaps even trivially so, explanation of why it's precisely e^-x^2 and not something else: Something in nature tends to be the sum of random parameters, and the sum of random parameter tends to follow that curve. (The more random parameters are summed to the total, the more closely the curve will approach that function.)
super awesome, thank you guys it was eye opening
What's the name of the music that pops up in the background at 3:46
Somebody help me
Resonance, by Vince Rubinetti: open.spotify.com/track/7KKRUp4psPm5ERKdqcBiyE?si=c396dffe7f2b470d
@@3blue1brown Thank you!
I love math because it is useful and accurate, and my friend loves it because a single spark of genius, usually seemingly unrelated and random, could make things a lot easier. Needless to say, the fact you answered these questions in the order that would be explained in your example conversation amazed us both
What a shame you've resorted to ai art. good math tho
in case this comment somehow actually gets seen and people cry about me dissing ai art: the issue isnt whether its "real" art or not, you can't define what "real" art is in a way that encompasses all art and excludes all not-art. the issue is how it gets its data, usually by stealing artists' work without their permission or knowledge.
Never thought I would get cliff hanged by 3b1b 2 times in a row!
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 ❣ .
I absolutely love this channel!
I have been teaching this derivation of the volume of a D-dimensional sphere for years, it is needed in statistical mechanics, when deriving the properties of the perfect gas within the microcanonical ensemble. As a side product, one can show that the ratio between the volume of a shell of ARBITRARILY small (but FIXED) thickness and the volume of the ENTIRE sphere tends to 1 as the dimension D tends to infinity (the volume of an infinite dimensional orange is concentrated in its peel)
It is such a delight to watch these videos... Really love how you bring the intuition of Pi into the minds of the uninitiated...
BTW, 'Pi Day' is just around the corner. Our condo management is organizing a 'Pie party' (of course at 3:14 PM) - how sad!! (this is somewhat like celebrating Gandhi's birthday with a candy party..:))
Would be great if you can suggest some real activities for the day that respects this amazing number...
For the animation at 21:10 - 21:30, why do the corners* start to shoot up before c becomes positive?
For all c0 the corners* should shoot up to infinity….
*I know the graph extends out infinitely far from the center - by “corners” I mean the corners of the section of the graph that are visible in the animation.
Your videos are mesmerizing
I wish my teacher would have found such ways to teach me
Dear 3Blue1Brown, I'm a fan of your contents but I rarely comment.
My comment here is to bring your attention to a potential mistake in this video production of yours.
I'm referring to the part containing AI-generated pictures.
I immediately spotted them as AI-generated, because they are full of artifacts: distorted and unnatural faces for the two ladies; ugly hands; cadaver-pale colors in Maxwell's face; hard edges everywhere; VERY ugly pixelization and JPEG-like artifacts all around.
This feels cheap. Very cheap.
And this uncomfortable feeling is inevitably impacting the viewer's perception of your content. And, as a consequence, the sense of worthiness attributed to your work.
There is a reason why good artwork is costly and time-consuming.
And, if you really appreciate the value that artworks can bring to your contents, that's exactly why you shall consider hiring a proper artist.
Don't fall in the cheapness trap. Safeguard your contents.
[Disclaimer: my PhD thesis is about ML architectures, so I'm well aware of AI limitations.]
Hi Federico, I hear your concern. For what it's worth, the goal certainly wasn't to be cheap; I did hire an artist to take care of those scenes, and we decided to experiment with using Midjourney in the spirit of better engaging with the full scope of tools out there, though the artist still had a very active hand in deciding on the overall concept and refining what Midjourney offered as a starting point.
My opinion on AI art is still not fully formed, and I appreciate your feedback. I'm curious, do you have an objection to the premise of using these tools at all, or if the artifacts were sufficiently addressed by the artist, would that be enough for it to feel alright?
@@3blue1brown My opinion on AI art is still not fully formed either...
We all know that those visual artifacts will surely progressively diminish with time.
For now, I think they are just ugly and distracting (and feeling "cheap and quick", as I told you in the previous comment).
Nonetheless, I believe that untill there is a serious law framework regarding the copyright issue, many people out there will see the use of those tools as a sort of "theft". And there are many artists in the science community (your viewers)...
I'm not an artist, so I cannot speak for them, but I can surely see their point.
Regarding me personally, I really like when things are well made: with intelligence, vision and taste.
Being it a gorgeous oil painting or a insightful mathematical video, it's like having the possibility to breath fully and profoundly.
I think the aim should be to amaze and delight, chasing the best.
One of the best videos, many thanks from the Netherlands
I like how your videos push my brain to the verge of meltdown without going over the edge.
Why the heck are you so smart. I don't get it. How do you figure out the most intuitive and insightful ways to explain things? You're incredible Grant. I'm baffled. Not by the video (your explanation makes complete sense), but by you in general.
What a tribute and visualisation to probably my all time favourite integral! Wow!
미쳤네...
Your video s are so beautiful and and the visual concepts are so illuminating , that almost every single time i forget to give a like .... and i think this is true for many viewers of your creation ...
Beautiful artistry! Well done
I had a VISCERAL reaction to the announcement that this would have another video. I was sitting there getting by pumped up for the answer about the two dimensions thing and then when he said I wasn’t going to get it, I reacted like a four year old who was told he couldn’t open his Christmas presents. Literal tantrum.
10/10 Would Recommend.
Your work is becoming of cinematographic quality. I would go to the cinema to watch this.
We think about Pi as the ratio of circumference to diameter because that is the simplest example and we learn it the earliest. In reality Pi has to do with continuous change - a circle is just a line that is continuously turning in on itself at a constant rate.
Stern-Brocot tree has the nice property that when we transform the fraction a/b to "simplicity" 1/ab, then all simplicities of each new row add up to one.
Denominators of the simplicities have natural interpretation as parallelograms aka outer products.
When we further interprete Stern-Brocot rows as winding numbers with circular curvature increasing from row to row, and keeping in mind that the density of fractions drops dyadically between whole number intervals, there seems to be much less need for probability theory approach...