Researchers thought this was a bug (Borwein integrals)

A pattern of integrals that all equal pi...until they don't.

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

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Hindi: Pragna1991

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Original paper from David and Jonathan Borwein

Other fun coverage of the topic:

Correction: 4:12 The top line should not be there, as that integral diverges

Timestamps
0:00 - The pattern
4:45 - Moving average analogy
10:41 - High-level overview of the connection
16:14 - What's coming up next

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

You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

Download the music on Bandcamp:

Stream the music on Spotify:

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Various social media stuffs:

[smartereveryday]

This is amazing.
I even love the way you visually explained moving averages.

[marshallmykietyshyn]

One of the main problems I have in making presentations is that I always try to make them like a story, avoiding spoilers so that everything leads up to the interesting take-home point, but you don't know what is coming until I get to it. This channel demonstrates why that's a flawed way of thinking for educational purposes. It's so much easier to follow along with these explanations knowing where they are going. The explanation at 4:22, while seeming like spoilers to me in the moment, was actually extremely helpful.

[brianparisien]

I'm a retired electro-geek who last studied this stuff over 40 years ago. Having just discovered this channel, I wish I'd had this resource prior to slogging through the computational mechanisms available to us at that time. These verbal and graphical explanations are absolutely fabulous, and I foresee hours of enjoyable education in my future with a cup of coffee in one hand, these videos on my side screen, and a spreadsheet in front of me. Thank-you!

[istvankertesz]

fun fact: Bor means wine in Hungarian, and Wein means wine in German, so if you translate it, it's the winewine integral.

[MDA.]

For the very first time, the bug actually WAS a feature

[diarya]

"a tiny positive number my computer couldn't compute in a reasonable amount of time"

You should do it the Matt Parker way, put it up on the internet and people will improve your code by a factor of millions in a matter of days!

[Pilchard]

I've been trying to wrap my head around convolutions forever, so seeing that you're going to be doing a video about them has just made my day :)

[harrywang]

So if we alter the series with 1, 1/2, 1/4, 1/8, 1/16… the integral will always be pi since the sum of this series will always be less than 2

[johnreid]

I’m currently studying maths at undergrad level, and the difference between 3B1B and the teaching I am receiving is day and night. You do so much to motivate and illuminate with these videos. I know that to learn the detail will involve a lot of hard work, and then I’ll have to develop my understanding by exercises and problem solving. However, now that I am fascinated and have a picture, this is a joy, not a chore. Thank you so much and keep doing this sort of thing.

[lemonlordminecraft]

Hey 3B1B team and especially Mr Sanderson,

I just wanted to say your videos never fail to enthrall and impress me. You have such a way of communicating high-level concepts that makes me feel exceptionally well-informed about the subject matter you cover. As of 3 days ago, I've finished my Bachelor of Mathematics degree, 4 years after having my love of mathematics reinforced by your popular video about 4 points on a sphere.

Your channel and its content are so important for young, mathematically-interested people and I cannot express how grateful I am for this content.

In so many words, thank you.

[Kyurem_originale_Form]

As an electrical engineer student as soon as I saw sinc(x) I immediately thought: Ah yes, definitely something with Fourier Transformation later in this video. Here we go again!

[johanneseller]

As someone who worked extensively with convolutions and Fourier Transforms in physics and engineering: This is a beautiful video and I’m excited to see where it leads us.

[faluffel]

Is it weird if I'm not studying or doing anything remotely to do with this kind of math, but absolutely loved it? It's strangely soothing and entertaining.

[SUNRA]

I'm a retired machinist and I ran into this twice this while machining radii from for example 9.500" to 8.500" in decrements of .01". I called tech support and no one knew the answer to this. They had never heard of it. Now I know, 15 years later.

[trevorbradley]

I was taught by both Borwein brothers (Johnathan and Peter) at Simon Fraser University in math undergraduate here in British Columbia, Canada. Peter was a joy to take complex analysis with. Jonathan's 4th year real analysis course was... less joyful. Brilliant man, we as his students weren't ready to hold the volumous and requisite knowledge in our brains at all times. Still, I greatly appreciate the experience and am glad I passed his course!

[thecanmanification]

i’m currently studying electronic engineering and i’m pretty familiar with all of this frequency domain stuff, but the sudden “aha” moment I had at the end was really something else. 3B1B really knows how to neatly wrap together seemingly disparate pieces of information

[bregottmannen]

Its so nice when you know enough math that you can figure out the problem yourself midway through the video

[tamashellwig]

As a Hungarian-German, the name Borwein is pretty funny:
Bor in Hungarian translates to wine, and so does Wein in German. So their name is basically wine-wine

[Scott-wdcq]

These video's are so incredibly well made that, not only is the math beautiful and well-explained, but the scripts 3Blue1Brown uses in these videos is just as beautiful and meticulously constructed. This is one of those subtle things I love about science and math - that it teaches you to speak carefully such that what you say has exactly one meaning. It's a truly difficult art to master but if achieved, the speaker is effortlessly satisfying to listen to.

[ahmedhani]

For some reason watching this video the though of DNA telomeres jumped into my mind. The fact that they shorten but remain relatively functional all the way until that critical threshold after which they fail to produce coding sequence protection. It’s just fascinating how our world’s laws just mesh and meld into one another from math to biology to space-time geometry