Pi hiding in prime regularities

A story of pi, primes, complex numbers, and how number theory braids them together.
An equally valuable form of support is to share the videos.

Music by Vince Rubinetti:

Timestamps
0:00 - Introduction
1:39 - Counting lattice points
5:47 - Gaussian integers
10:30 - The lattice point recipe
17:50 - Counting on one ring
20:14 - Exploiting prime regularity
25:19 - Combining the rings
28:36 - Branches of number theory

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

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[rgbatom]

Pi is hiding everywhere. This is a prime example

[yerivalpolanco]

Being able to watch this kind of content so easily and for free is probably the best thing about living on this time.

[adjoint_functor]

26:37 genuinely yelled in shock and awe at this part. This right here is why I love math: those moments where all the complexity and intricate patterns collide and combine into one clean and beautiful whole. This video made me more excited and intrigued than most movies. Bravo, Grant, bravo.

[Leyrann]

26:34 The most beautiful thing in the world is when half an hour of complicated math comes together in something so simple and direct.

[danielc]

This method of teaching, it's a revolution in pedagogy really. A calm warm voice explaining the hidden structure of number theory, with a detailed and colourful animation that is perfectly in sync. I've never seen anything quite like this channel.

[MrDaanjanssen]

23:50: 'Make sure everything feels good up to this point'

*Starts the video all over again for the 4th time*

[AnyVideo]

Shortest 30 minute video I've seen in a while. I just cannot believe how high quality your videos are.

[ahoustonpsych]

Hey I haven't commented before but I just wanted to say that I am absolutely in love with what you're doing. It's clear how passionate you are about this stuff; your passion and the care you put into these videos an extreme pleasure to watch. The animations are downright astonishing in how well you've managed to make all the right visual connections to complement the explanations that you're giving. Not to mention the extremely high quality of the animations is far above what I've seen anywhere else. Your voice is very calming and you speak very clearly in an approachable and inviting way which really helps hit the nail on the head. There's no question that you understand what you're saying at a deep level. Really everything is freaking awesome and dead-on.

I can't really express the impact you've had and are continuing to have on me, especially as of late. I had been passionately studying machine learning over the last few months, putting almost every bit of my free time into reading something about the software or the math involved within. It occurred to me recently that I need to go back to college to learn some of the high-level stuff so I had been specifically studying into basic calculus for the placement test. I was floored when essence of calculus came out, and doubly so when I saw that you're looking to do a series on probability which is heavily involved in machine learning. This excites me in ways that I haven't felt before, and has been adding fuel to a deep passion for this stuff. Not to mention other videos like these ones which goes deep into subjects that you definitely wouldn't expect to see.

It sounds a bit dramatic and a bit rambley now that I actually type it all out, but I am serious. It's hard to truly put it into words which is why I'm glad you have a patreon. Everything down to the quality of your ads is top notch (which is actually what prompted me to post this since I'm checking out remix).

This was all super gross but I just had to let you know how much I appreciate you and what you do, and how much I enjoy the content that you're making here.

[smogy]

8:16 "this might seem needlessly complex"
I see what you did there

[matthewkellman]

This proof and the explanation was truly incredibly elegant. Thank you for putting in the time to create these fantastic videos. Your channel is truly unparalleled in this universe!

[Benny_Blue]

I never in my *life* thought I’d understand where that pi infinite series came from. This video is spectacular — I followed along the whole time. Then afterwards, although it was tough, I successfully redid all the steps mentally, and they made perfect sense. Thank you so much for this — you did a great job!

[SerranoAcademy]

You, sir, are an absolute genius. Thank you for what you're doing.

[DEWILL]

Seriously, we don't deserve this quality of explanation. And we cannot express how grateful these videos are.

You deserve my loudest clap ever. Thank you.

[rohanshah]

I am just startled by the way you put together concepts of Gaussian Integers, Prime regularities, and Multiplicative functions to obtain such an amazing result. Was just amazed to see all these boil down to such a beautiful equation. Thank you so much for making this video. I really appreciate your efforts to bring such complex yet beautiful results to math enthusiasts like me. Thank you very much.

[AroundTheBlockAgain]

Was NOT expecting the "hey why don't we sort these into columns" at the end but D U D E
That was BEAUTIFUL

I thought you were going to pull a fancy "and here's how to add up every integer intersection on every single circle this radius or lower" but that was even wilder than I was expecting. Super fun ride. Math Good.

[MrDaanjanssen]

3Blue1Brown, you should know that someone just stood up from his chair and gave you a standing ovation

[tomconti-leslie]

Wow. Like seriously, wow. I've been watching your videos for a while and I've seen them getting longer and longer, but as I saw someone else say in the comments, this is the "shortest 30 minute video I've seen in a while"! Please keep going with these sorts of videos, they are totally worth it. The visual aspect is absolutely perfect, and that beautiful coming together of everything you'd spent the last 20 minutes setting up at the end of the video was just amazing - I started grinning around 26:50 when I saw it all happening ;) I should probably be studying for my exams instead of watching this but this is so much more rewarding... Thank you so much!

[anselmschueler]

"This might seem *needlessly complex*" Nice joke there!
(8:15)

[kjyhh]

0:11 Wow, the PI guy jumped into the formula.

[nUrnxvmhTEuU]

7:47 "They're called Gaussian integers, named after Martin Sheen“ :D