There's always a moment in these videos where I suddenly realise "oh, I can see where this is heading now", and I immediately grasp how a bunch of mathematical ideas I thought had nothing to do with each other are actually working closely together. That moment never fails to bring a broad smile to my face.

What's better than a 3Blue1Brown video? A 3Blue1Brown video with an original proof by the creators

Novel mathematical proofs being presented on KZfaq is proof to me that we live in the future.

A mathematical proof first shown on youtube by the best mathematical youtube channel ... What a time to be alive !!

You're a freaking math communication genius dude

Even though the argument given here is new, the Wallis product has been known a long time, and there are other arguments for it than the one given here. For example, just as our previous video on 1 + ¼ + 1/9 + … was based on a paper by Johan Wästlund clearly showing the connection between that sum and circles, there is also a beautiful paper by Wästlund showing a connection between the Wallis product and circles via a different approach than we’ve taken here, which you may find interesting. Donald Knuth has also put out descriptions building off this work by Wästlund. You can check both of those out in the links below. And of course there’s Wallis’s original 17th century argument, based on analysis of certain integrals, though this can make the connection to circles hard to see directly. But, naturally, we’re fondest of the proof we ended up giving here, for its simplicity, for the directions in which it generalizes, and, hell, for the opportunity to re-use our lighthouse animations. And we hope you enjoyed it too. *Edit*: It looks like some people are asking about why the segment at 12:33 is okay, given that it feels like taking 0/0. Keep in mind, the actual goal at that spot is to find a polynomial whose roots are L_1, L_2, ... L_{N-1}, so the concrete result being stated is that (x - L_1)(x - L_2)...(x - L_{N-1}) will expand out to become 1+x+x^2+....x^{N-1}. No division by zero issues there. Sure, plugging in x=1 to (x^N - 1)/(x - 1) is undefined (at least before explicitly stating the intention to extend the function via a limit), but the reason for doing that polynomial division was just to see how (x - L_1)(x - L_2)...(x - L_{N-1}) would expand. All that division is asking is (x - 1)(...what?...) = (x^N - 1). Here, to give a really simple example, it's like saying x^2 - 1 has roots at 1 and -1, so dividing it by (x - 1) gives a polynomial with just a root at -1, namely (x^2 - 1) / (x - 1) = x + 1. "But wait!", someone could say, "you can't plug x = 1 into that fraction!". For sure for sure dude, but that doesn't change the fact that x + 1 is legitimately a polynomial which just has -1 as a root. Maybe you justify that division by saying something about limits, or about analytic continuation, or just by reframing to say what you care about is the question (x - 1)(...what?...) = x^2 - 1, but that's all kind of beside the point. Also, many of you are asking "Isn't the 'distance is proportional to angle' approximation only valid for lighthouses near the observers? What about all the lighthouses on the far end of the circle?". The key is that the product we are ultimately interested in is made up of the asymptotic contributions of each particular lighthouse (in the sense of, e.g., "The 53rd lighthouse after the keeper"), in the limit as N goes to infinity. Whatever particular lighthouse you are looking at, in that limit as N goes to infinity, it will be bunched right next to the observers, and so distances will be proportional to angles for computing its asymptotic contribution. As noted in the section on formalities, Dominated Convergence then rigorously assures us that it's ok to equate "The product of each particular lighthouse's asymptotic limit contribution" (which is the product we're interested in: the Wallis product, or sine product more generally) with "The asymptotic limit of the product of the contributions from each particular lighthouse" (which is the asymptotic limit of the products we have an easy time calculating: the distance-products our lemmas directly address). For more technical details on this use of Dominated Convergence, see the supplemental blogpost. Our supplemental blogpost: www.3blue1brown.com/sridhars-corner/2018/4/17/wallis-product-supplement-dominated-convergence Another cool way of approaching the Wallis product: www.math.chalmers.se/~wastlund/monthly.pdf apetresc.wordpress.com/2010/12/28/knuths-why-pi-talk-at-stanford-part-1/

The sheer quality of this content is quickly surpassing not only everything else but itself as well. When I thought 3b1b couldn't get any better it does twofold yet again.

Woooow!! Congrats for the beautiful proof! This is both extremely cool and hugely impressive!

I keep noticing, how many interesting ideas come out when you connect square-related and circle-related concepts!

Your videos are the most creative representations of mathematic principals I’ve ever seen, it’s always such a joy sitting down to watch one of these. You’ve turned math back into the beautiful and elegant process that I lost touch with, please never stop doing what you do

Every time I watch one of your videos I feel the urge to tell you how awesome your channel is. This connection between the roots of unity and the product expansions of sine is something I didn't know yet, explained in such an intuitive gemetric way! Seriously. Thanks for doing all this!

TAU SPOTTED

Sridhar Ramesh! That guy's writing on Quora is pretty awesome, good to see he's joined the 3B1B team. Great proof, and I'm proud that I caught the technicalities myself; was going to write a comment about them, but as usual, you have addressed them yourself!

I love the little introductions before each video. It's interesting to hear some background on why exactly you made it. It's always an amazing feeling when you stumbled upon something new like in this one!

I always have to rewind and rewatch your videos so that I’m sure I have really grasped everything that is going on. I’ve been watching your videos for a few years now and I love them!

I, as a 10th year student, haven't understood anything, but that give me more interest in studying maths! So, great job! It means that your videos are really interesting and that you're encouraging more young people like me to study the beauties of maths!😀

You're a genius educator, I mean it. Every single one of these videos is so carefully made. Even though I'm already familiar with most of the concepts I find myself learning. It really blows my mind. I just wanted to say that ^^ Congratulations for the proof!

The Wallis product looks very musical to me. 2 is going up an octave, 2/3 is going down a perfect fifth, 4/3 up a fourth, and so on. I don't know if it's possible, but it would be interesting if there's some kind of proof from this direction. Basically, the Wallis product gives a series of musical intervals that converges to a "note" that is pi/2 above the starting note, which works out to about 782 cents, a rather flattened minor sixth.

Noo! That poor Pi creature at 3:03 became a skeleton!

I'm blown away with every new video. I said it before and I say it again : this channel is pure youtube gold ! Thank you, I genuinely thank you.

I was looking for a way to "generate" pi out of this product from scratch, so to speak, but this is the next best thing I could have possibly asked for. Your use of infinitely large circular lakes with lighthouses, observers, and light-reception-metrics to prove facts about infinite products and sums is by a significant margin the most beautiful mathematics I have ever (and probably will ever) encounter. I would love to see more of this kind of thing, even if its another video explaining another set of previously known results.

This is really cool, and that is not even my favorite video of yours. As you mention in the beginning, a big part of the value in your videos is attributable to the presentation and the communication of the result and I would like to say that you do a fantastic work in this matter. The representations you propose are always very insightful on top of beeing beautiful. In essence, thank you for the hard work and keep doing such intersesting an wonderful videos ! (Sorry for the approximative english, hope you still get the message =) )

I love how you lead me to an understanding. When you teach, I grok. The way you build up clues for me to start piecing things together is akin to being lead through the plot of a great mystery novel. You first help one to construct an intuition and then you reinforce it. This wonderfully developed skill-set you weild shows the beauty of your mind. Because of you (and a few other brilliant minds) I am able now to learn anything mathematical if I just think about it geometrically/trigonometrically. Array manipulation perceived as translation and rotation through 'N' Dimensional space has changed the way I see the world. I love you for this. Thank you for sharing your understandings in such a beautiful way!

I’m a simple man, I see 3B1B, I click like.

Who else had no idea what he's actually talking about but can't stop watching his videos

Mathologer and 3Blue1Brown are honestly legends, revolutionaries. You guys change the world with every video. Absolutely amazing communicators, a skill sadly rare in higher education and complex topics. decades from now, you guys will be like the Feynman of math education. Keep up the amazing work

The vibes of the start of your videos are amazing! Always encourage us to be curious about mathematics.

It's amazing how your videos keep getting better and better!

"local mathematicians" i don't think my town has any

The level that I love this channel, cannot be expressed with real or imaginary numbers. I am 40 and a 3rd try 1st year Calculus student but the amazing thing, to me at least. Is that there are small snippets of videos like this, that I understand or look "familiar" to me in some way. I view that as my own personal mathematical enlightenment developing. Honestly, that makes me love maths even more; even though I struggle horribly with it. I have a brilliant calculus teacher who thankfully seems to have patience with me as well as the host of tutors that help me every week. I hope one day, I can look back on these videos and either expound upon them, or say "Ahh, yes. It is that way" and actually understand why. Thank you Grant and everyone that contributes to these visual dialog, and that includes people in the comments who push questions with more questions.

everytime i watch a 3b1b video i always manage to get lost, then completely enlightened at the end. insane quality vids

Such a wonderful presentation of this concept, something that is so abstract is explained in such a lucid, pleasant, logical and visual manner ! Way to go ! I am a big fan. Binge watching math videos first time in life :D

My mind has never not been blown by this channel!

Great that you talked about the issue with limits and infinite products !

I love your voice. It helps me fall asleep. You’re not boring but rather quite calming

I always had a hard time convincing to myself the infinite product representation of sin z in complex analysis class. It all make sense to me now. Thank you so much!

Thanks ♥ for you efforts and for these amazing videos, Please we want another episodes about deep learning and Machine learning algorithms (RNN, K-means, Logistic regression, SVM/SVR, ....) ♥

I appreciate the comments about convergence of a product. I'm not an analyst, but I felt like you'd done something a bit naughty in the final steps of the proof.

I'm happy that firstly I had the time to watch the whole video at once and secondly it was that long. Keep it up. I keep recommending your channel.

A new 3b1b video on 4/20? You never fail us

24:47 "=" sign is missing in 1/1^2 + 1/2^2 + ... = pi^2/6

It is one of the most understandibale and captivatng explanation I've ever seen. Thank you!

Great Viedo (watched the unlised one), just thought I'l let you know you did a great job, as per usual. This one does take it to a new level tho :D

I recently saw your appearance on HFS with Hank Greene, I must say, you were an excellent guest! I'd love to see you collaborate with other channels more. Have a great day!

Math totally aside, damn your graphics are well done. The foggy effect around those LED-palette lighthouses against a night background really pulls me in for some reason. Your graphics are very good on your other vids too, but this one made me think to pause and comment on it.

This video is a lot harder to digest than your previous content. But that means I get to rewatch this video multiple times until I think I get it. Great animations though :-)

Great video as always. The fact, that I did not understand the "keeper/sailor" idea is my fault of lack of brain mass. A very complicate proof. The one shown in a Mathologer video is more intuitive and easy. But anyway, I like the lighthouse concept a lot! Brought much insight to me.

There comes a time when I'm not actually able to get what are you talkin' 'bout but still I watch it further..and I realize that I'm digesting it now!

8:14 The tau rebellion will never die

This is just.... hell, how did you get such ideas?! They're so convoluted I don't see a way for someone to 'notice them.'

Your mathematical visualisation left me with no words how to exactly admire them, keep up the good work Can i make a suggestion plz bring this material to virtual reality too, like the platform of oculus go .

Long time viewer first time commenter ;-) Thanks for the amazing math videos. I love to share them at work and at home. I was a little troubled by this proof when the distance ratios are near to halfway around the circle where the small angle approximation no longer holds. What am I missing?

Everything is linked. That's awesome! Euler would have loved this video:)

Your use of colour in your videos is always beautiful. Keep it up :)

Nice! I think you also almost demonstrated the gamma function reflection relation for the sine cardinal there.

I also notice that, taken in pairs, this product is prod[ n^2 / (n^2 -1) ] for n even, and is also 2 * prod[(n^2 -1) / n^2 ] for n odd. It seems that those forms should provide a relationship to the Basel problem and thus also to sines. The most interesting thing to me about Euler's solution to the Basel problem is that it only uses the coefficient of one of the powers of x in the expression for the sine.

I see that subtle easter eggy use of tau instead of pi to record the complex number's phase angle around the unit circle when talking about roots of unity. ;) Makes it so much easier to communicate radians relative to a complete turn of the unit circle~

I'm glad I watched enough mathologer videos to instinctively question the commuting of limits and interweaving of infinite products. While, I can't determine when those are possible, it's still nice to know I learned something.

I would freaking love to be so accurate on the graphics of my videos. Yours are so awesome... Very a great job you've done here !

Beautiful, thank you for sharing such a satisfactory result.

This is so good I feel equal parts joy and sadness. Joy over how great it is and sadness over how poorly math was communicated throughout all my years of studying it.

11:59, a fancy way of dividing by zero without the universe ending. :)

I wish a more standard mathematical proof was also provided alongside with the long paragraphs of discussion in the blog. Its good to provides insights on how the proofs were constructed, but there are still other technicalities underlying the proof. I know you guys defintely are capable of doing so. For example, another issue I am considering is that although all of the ratios are converging to its numerical limits, the speed of which they are converging is slower and slower going down the sequence, i.e. terms are not uniform convergence. I suspect that causes the interchange of the orders generate a different ''value'' for the limit, rather than "absolute convergence" you mentioned in the blog post.

Your music soothes my soul. my BP drops immediately and i feel happy.

This is the best channel on youtube!

3:03 when an extra gets a part in the movie

In all the videos I just think and at the end I realize how elegant the solution is.

"When all the algebraic dust settles..." Haha that is a great term I'm definitely using this soon. I am teaching surface integrals and there can be a lot of algebra involved with taking cross-products of vectors in, say, spherical coordinates. Funny term

Casually gonna demostrate de Wallis product of pi and the sin formula Mindblowing dude, just mindblowing. Don't be scared at all to explain more, and more slowly as well. I'm always afraid of losing details. I would even recommend you to make 2-parts videos, at least I ould recommend you consider doing it. I mean, you don't make such a beautiful proof everyday...

How can you dislike this guy?

Just came back from school, and I'm ready for more learning.

Very elegant. Thank you for your hard work!

3Blue1Brown: explains an exclusively arithmetic complex problem with a very specific solution Also 3Blue1Brown: so you're probably wondering how this relates to geometry

I used to think imaginary and complex numbers where something stupid. How wrong I was. Now I know that even the most abstract ideas in math can allow us to find very deep and interesting relations between things

3Blue1Brown I think you could improve the sound quality a lot by dialing out a bit of low end + low mids (the proximity effect of your mic) with a HPF & EQ; it's a bit boomy and makes it harder to understand as is. Anyway, love your work!

You are the most underrated channel on KZfaq. I’m sad that you don’t get millions of views despite making interesting videos.

I had to present a proof of the stirling formula last semester and the main part of it was proving the Wallis product (after you have this the rest is more or less trivial). I also had an geometrical approach based on a paper I found but it wasn't as visual as this here of course. I must say that I didn't like this as much as the Basel problem video because it was more algebraicly playing around than geometrical intuition. I also think think that this approach is a bit harder to make rigorous for a non-mathmatician audience. If I don't have a rigorous proof then I can't be sure that my intuition is correct. So this proof here is not ideal for my taste but I really like that you try to make "nice" proofs for already proven things :) EDIT: I just saw that the the sorce I used for my proof was the one by Johan Wästlund in the description. I of course rephrased a lot of stuff he did to make it easier to understand. I think his method is easier for everyone and could even be done in schools but the main problem is the proof is pretty boring in the middle when deriving a lot of side stuff algebraicly.

Am I missing something? At 12:33 when O = 1, the fraction on the left becomes 0/0, so how can you still conclude that it equals a partial sum of the geometric series?

I don't want to imagine all the work put in !

This is an amazing video. Just one thing: I can't view this in HD because it's 60 fps and my device doesn't support that and KZfaq forces me to use 480p. I think if you upload at 30 fps more people can watch this at HD quality. In my opinion, 30 fps should be enough for nice animations. Resolution is much more important.

OH YES. Another 3b1b Video.

I like how some people dislike the video when it’s only been up for 2 minutes and the video is 25 minutes long...

Beautiful proof and beautiful video. Really a great job!

Im just doing my thesis on abelian number fields, hence lots of cyclotomic fields stuff is in play (thanks to Kronecker-Weber). I wouldn't be surprised if this was burried somewhere in the bast literature, but it is cool as heck. I would love to see more.

I think it wouldn't hurt to quickly mention that the absolute value of a product is the product of absolute values also when talking about complex numbers

19:55 I didn't get that, why is the limit of the collumns 1? :/

Amazing amazing video! Thank you Sweether and Grant!

Thanks for making these videos.

Great video again! Just one small suggestion. I found 9:00 quite confusing at first, as I thought the distance is not simply (O-L_i) but |O-L_i|, and the magnitude of a multiplication of the former is not a multiplication of the latter. Then I did search around and found the property for complex numbers that |xy|=|x||y|. Maybe it would be helpful to note this property somewhere for people who are not that familiar w/ complex numbers.

Notification squad here. Been waiting for the video to come out :)

And like every other 3b1b video, THIS WAS MIND-BLOWING

I recently had to prove these things in a home assignment. You start with the Fourier series of cos(tx), then use it to show that the integral from 0 to xpi of [cot(t) - 1/t] is the infinite sum of log[1 - x^2/n^2] But doing standard integration you also get log(sinx/x) So that means log(sinx/x) = sum of log(1 - x^2/(npi)^2) = log of the infinite product (1 - x^2/(npi)^2); You get rid of the log in both sides and substitute x = pi/2... voila

8:18 LONG LIVE TAU!

I'm in love 😍 with 3B1B

Amazing doesn´t describe this accurately, this project is far beyond that, keep it up! At 12:51 you can also show that the left side of this equation exists, when you substitute x with the value x=1 by using L´Hospitals rule, so lim x-->1 of (x^n-1)/(x-1) = lim x-->1 of (n*x^(n-1))/1 = n. Absolutely brilliant stuff in my eyes!

Great content my friend.. I wish I discovered your channel a bit sooner.

Why is (at 19:52) the limit of 7*1*1*1*1...=1? That doesn't make sense to me.

Man this sounds interesting, but your animations are too good to be watched in 360p, so brb.

Man, you do a very good job in bringing down math to an intuitive level! It's interesting, how math reveals all the suttle relationships between certain things, that seen to be totally separate from each other. Even with things in the real world. Like circles and infinite sums and products and their connection to physical quantities. Now, there still remains a question to me: x^N - 1 = (x - L1) ... (x - L2) was justified by saying, that both are equal to 0 at all the points x = L_{0 to N}. But how then can you generalize that to any point x on the complex plane for every N?

How dare you make something so complicated so clear and simple using clever techniques and beautiful animations. Outrageous. Who do you think you are?