Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

A curious pattern, approximations for pi, and prime distributions.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/spiral-thanks
Based on this Math Stack Exchange post:
math.stackexchange.com/questi...
Want to learn more about rational approximations? See this Mathologer video.
• Infinite fractions and...
Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video:
• Prime Spirals - Number...
Dirichlet's paper:
arxiv.org/pdf/0808.1408.pdf
Timestamps:
0:00 - The spiral mystery
3:35 - Non-prime spirals
6:10 - Residue classes
7:20 - Why the galactic spirals
9:30 - Euler’s totient function
10:28 - The larger scale
14:45 - Dirichlet’s theorem
20:26 - Why care?
Corrections:
18:30: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence.
Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this, but I could be wrong there.
In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann!
My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
These animations are largely made using manim, a scrappy open-source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@ktu6133]

At this point, the word “beautiful” isn’t even enough to describe the sheer elegance and clarity of these videos. Amazing as always.

[@petemurphy7164]

Yep!!!

[@DoloresUmbridgeairsoft]

i was just about to say beautiful

[@shitalsavekar7757]

Couldn't agree more!!!

[@STriderFIN77]

also Amazingk!!

[@jalilcompaore]

How about bootyful?

[@Supertimegamingify]

It's clearly not pointless, I mean, look at all of those dots!

[@samuelbarham8483]

I feel this comment is underrated.

[@aryankumarprasad1574]

@@juhonuorala3512 That's a nice line.

[@user-oc7mk7qe6k]

lmfao

[@cordongrouch9323]

But even a dot has an inside and an outside. Of course, that is not the point of zero dimensions.

[@minecrafting_il]

you now have 0 likes in 8 bit systems.

[@bendahou3778]

As a maths lover, proving a theorem before you knew it existed is undeniably the best feeling I would ever experience

[@DonkoXI]

It's funny how the things you enjoy change when they become your job. For me as a mathematician, proving a theorem only to find out it's already been proven is frustrating. It's not entirely bad, because at least the fact that it's been done already means your proof (probably) isn't wrong. You also walk out of it understanding things very well, so it's not a waste of time. It's just frustrating that you can't turn your work into a paper (unless your proof is very different, in which case it's sometimes still worth publishing).

[@aureliontroll2341]

I remenber when i make the area formula for the diagonal of a square based on its side ( diagonal = sqrt of 2 Side) when i was at high school learning sen and cos , i was so freaking happy that i made a formula that give the awnser for common problems. Only to discover a year (?)later that that formula already exists.

[@AwakeAgainAtLast]

Math is already plural. You don't need to add an s to say "maths", it's redundant.

[@DonkoXI]

@@AwakeAgainAtLast This is true in American English, but the convention is different in other countries. It's not a mistake, it's just a regional difference.

[@Solutra]

@@AwakeAgainAtLast woke up on the wrong side of the bed?

[@ralphengland8559]

My favorite approximation for pi is 977/311 because both numbers are themselves prime and have analogous locations when typed out on a standard number pad.

[@lucasb.bahadir7433]

That's actually really cool

[@guilhermeottoni1367]

Mine is always when you calculate (355+22)/(113+7) = 377/120 = 3.14166666... The repeating part has only one digit.

[@semen_tv8478]

977-311=666

[@JrgenHelland00]

@@guilhermeottoni1367 at that point, why not just write 3.1416? The rest of the sixes only take you further from the true constant while also being more key presses and a division.

[@baukepoelsma]

That's the most nerdy thing i ever heard anyone say, and i like it. The 977-311=666 makes it even better xD

[@avimohan6594]

"I had never heard this before but I find it too delightful not to tell." This dude's love for teaching is *SO OBVIOUS* and deep and genuine. Every video is made with special care and I won't be surprised if he edits each lesson about 20 times before uploading to get it just right. The *delight* is *ours,* Sensei.

[@chuksjr.1440]

Your reply is so apt and true. I want to teach like him.

[@CptPatch]

I've been working through the lectures he did for Khan Academy for multivariable calculus and he just has an amazing method of conveying the intuition of a concept visually before teaching the proof. It isn't as refined as his more recent work on KZbin, but I really appreciate what Grant does.

[@ancbi]

When I find something to delightful not to tell, some people around me just say "does it sell?".

[@lordx4641]

I always thought spirals r underrated hemchandra nos (popularly known as fibonacci no) himself showed the unique characteristics of spirals in nature let it be galaxies or flowers thats why the cholas had temples arranged according to golden ratio and golden mean

[@amusa8448]

totally agree... subscribed right a way

[@arpandhatt6011]

3Blue1Brown: Zooming out KZbin Compression: Dies

[@SARGAMESH]

@3blue1brown What an amazing video would it be if you found out how the nature of these primes shown on screen interact with the compression algorithm meta-ly to the video causing the algo to glitch out like that

[@arpandhatt6011]

@@SARGAMESH The reason why the video “glitches” when he zooms is that the video has a certain maximum bitrate. KZbin’s compression algorithm will not update pixels that don’t change. That saves bandwidth. However, when too much stuff is changing it has to reduce the resolution. There’s an amazing video on this. Google “tom scott youtube compression”. His video title is something about snow/confetti.

[@the.abhiram.r]

Arpan Dhatt mkbhd proved it with his 1000 upload test

[@Gabriel-sq6vy]

@@arpandhatt6011 Not to mention aliasing, which you'll unavoidably have at that kind of graphics

[@HermanWillems]

They should make the compression based on prime numbers, maybe that will specificly make this video better. But anyway, C++ is better than Rust. Just want to let you know aswell.

[@DelandaBaudLacanian]

3:22 "If you patiently went through each ray" I can hear it in your voice, thank you 3Blue1Brown for your meticulous work in counting each ray

[@teenyfroog6851]

Well he can just use the theorem to see that theres 280

[@EngRMP]

OMG, mankind is so lucky to have these two things: someone who can clearly explain some of the most complex subjects in math; and a simple means of making that knowledge accessible (KZbin). I don't mean to imply that producing these videos is "simple".... no, it takes A LOT of time and effort to produce a video this wonderfully clear. Who ever thought that when KZbin started, we would get to this point... we are so lucky.

[@rayahdesu1251]

Hello! I'm currently taking a Mathematics course in college, and I'm kind of questioning myself why did I even enter this course. This video made me realize why I love math, and why I entered a Math course in the first place. Thank you very much for these super high quality videos!

[@_stockfootage]

@@dsdsspp7130 ?! That's not math at all and college math isn't that either. Math at the university level is seldom about memorizing formulas but rather about finding the right solutions to diverse problems and showing how.

[@somedude4122]

@@_stockfootage Depends on the country.

[@haku1145]

@@dsdsspp7130 In my case it's all about demonstrations as of now. Knowing things like integrating or multiplying matrices is taught very quickly and isn't given much importance in homeworks/exams (most times, at least) compared to knowing how to demonstrate stuff.

[@MrMctastics]

unite perry at least take a proofs class. That's where math gets fun

[@thebluegaming7706]

Keep going!! Math is pretty cool.

[@aubreystewart3772]

I wasn't ready for how beautiful the "zoom out" was going to be

[@freewyvern707]

it loses so much after first viewing but is still brilliant

[@Lit_NightSky]

When he zoomed out all I could do was to stare at it, fascinated with my mouth open.

[@TheHwiwonKim]

Check the original answer from the link above. Once zoom out will shock you more. Because the beams are actually spiral again.

[@nikey2110]

@@TheHwiwonKim do they turen abck into beams?

[@danielwanger5919]

Time Stamp?

[@SechristCircus]

Hands down one of the best "math-y" videos I've seen. One of the best concept breakdowns as well. Everything is clearly described in an easy-to-understand way, yet you don't shy from all the "overly pretentious" (lol) jargon. Finally, the call to study and understand interesting concepts ("be playful") where you may connect the dots later down the road is the best. Thank you

[@nunyabidness6323]

What you said toward the end about accidentally rediscovering things people learned in the past bringing an intrinsic value to them that simply being taught lacks was...completely true. It reminds me of this time once in which I tried to use multidimensional arrays to represent the possible results of a series of coinflips and accidentally discovered that the number of heads has pascal's triangle embedded into it.

[@aeiouaeiouaeiou]

i just thought to myself: "wow this is fascinating. i cant believe i didnt know" but then saw that i actually already liked this video. it fucking sucks to be stupid

[@evelynwang2225]

LMFAO

[@Uyhn26]

LMAO RIP.

[@aeiouaeiouaeiou]

@@hybmnzz2658 heroin or hero in? x)

[@minecrafting_il]

@@aeiouaeiouaeiou hero(br)in(e)?

[@travisperry4515]

@@aeiouaeiouaeiou clever junkie

[@sorio99]

I’ll be real, seeing the switch of the spiral from clockwise to counter clockwise when we move from mod 6 to mod 44 is super satisfying.

[@GetIntoItDuhh]

I don't even LIKE math, but this was amazing.... and I wasn't completely lost for most of the video! You're a brilliant communicator.

[@shardator]

Math is like beer. You won't like it to begin with, but drink some good... And you are lost to it :)

[@GetIntoItDuhh]

@@shardator ive worked in a math-focused field for almost a decade; still hate it.

[@shardator]

@@GetIntoItDuhh applied math is not that fun. It needs you to focus on things you are not interested in. I'm a SWE, but hate programming, when I have to work on shit someone else wrote 15 years ago, probably drunk.

[@flamable2596]

Me: *a math disliker* (20-1 kicked my ass cus my teacher sucked) Also me: PATTERNSSS PATTERNS PATTERNS!!

[@johnchessant3012]

"in case this is too clear for the reader" lmao Also, I absolutely love the ending. 3b1b, I wouldn't be half as enthusiastic about maths without your videos. Thanks so much!

[@henryg.8762]

who would

[@odedzrubavel874]

@@henryg.8762 I would, and many others as well, but we sure do appreciate 3b1b for making amazing videos. It's people like him who make people that don't like math much at first, like it.

[@yto6095]

@@henryg.8762 i mean, i love math since i was 4 years old, and for the next 3 years i didn't even understand english enough to be able to understand any of this except that there's cool spirals and gaps between them. love for math is the kind of thing that you just need to start somehow, and then it grows on its own. it can grow faster, when you find things like googology or good math videos, or other stuff that is very enjoyable, but doesn't seem too useful at first, but even if you don't find these kinds of things, you can get just as far.

[@3blue1brown]

Important error correction: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence. Incidentally, his tactics also show that these residue classes have the same "density", but for an alternate formulation of density than the one shown in the video. Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this. In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann! My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know.

[@vari1535]

(first) How do you even make the visuals and graphs on your computer? Probably some programming or something :P

[@ArnaudMEURET]

Variety of Everything Our host spent a lot of time putting together his own rendering and animation platform. I hope he’ll give us a comprehensive tour one day.

[@juanluisclaure6485]

an important erratum and I surprise to myself get it in the second read. It is too specific information that is hard to google it and find some Wikipedia about it, well I don't research enough is true too. cheers!

[@Suav58]

@@wizedivine First, let's make it clear between ourselves, that plane is a surface of a 2-sphere with an infinite radius. Secondly: S1 sphere is a boundary of a 2d disc, S2 sphere is a surface of a 3d ball and S3 sphere is a surface of a 4d ball (neither of the latter two you can see, or, to remain on a side of caution, most of us can't see them). This goes on. I think, then, that you wanted to see something where spiral is drawn in 3 d space and coordinates are (r, α, β), where α and β are angles from the x and y positive axes. Pity we don't enjoy true 3d vision, but only a binocular ("stereographic"? - where did Riemann got his idea from) projection of such onto a part of a sphere. I guess, you can go on from here on your own. I think it would be doable in GeoGebra. (I think 3Blue1Brown should use the standard terminology for spheres in his other videos. Moreover, geometric algebra and a proper torus are waiting.)

[@LucaS-tf2sj]

3Blue1Brown I’m a German ninth grader and I like maths and ur vid but now my brain makes weird noises and smokes.

[@umaer009]

The excitement in your voice reflects the love you've got for mathematics. Hence, your videos are truly labour of love. KEEP IT UP!

[@marinanikolaou4585]

I took number theory the previous semester in uni and now i can see the point of it. Brilliant job! The visualization of such theoretical problems is so helpful

[@richardcarnegie777]

It’s always been amazing to me that early mathematicians could find the time to focus so deeply (without computers) on these abstract topics in number theory. Life then was generally shorter and rougher so they must have been incredibly dedicated.

[@WhyBhanshu]

On the contrary, one had a lot fewer distractions to lure them away from the thing that interested them. In this day and age of internet, it's very hard to keep yourself dedicated to one thing, there's always something else that demands your attention, that makes you feel like you're missing out on something.

[@AroundTheBlockAgain]

Yeah they usually had other people to do their cooking, cleaning, and errands for them. Life was shorter and rougher for their cooks, their maids, and other house staff, not them so much. ;)

[@General12th]

@@AroundTheBlockAgain Yes, this is true. The folks who figured this stuff out tended to be men of leisure, for whom day-to-day finances weren't a concern. Aside from the lack of modern amenities like electricity and running water, their lives were probably _easier_ than most of ours, not harder.

[@harshitkatiyar2250]

@@WhyBhanshu That's exactly what I was thinking last evening.

[@anarchodin]

"Life was shorter" is largely a myth, caused by interpreting the "average lifespan" too narrowly. If you exclude those who died before reaching five years of age, the figure jumps up. By a _lot_. The fact that infant and toddler mortality was high enough to have such a substantial impact on the average can be taken as an indicator that the second part of that statement is broadly correct, though.

[@SilverMustang920]

To begin with, I just can't even image how you even managed to make these stunning animations at such a large scale. ABSOLUTELY FANTASTIC!! Easily one of my favorite channels on KZbin.

[@katech6020]

He is using manim, which is a python library that he created to make this video. you can check it out in Github

[@TheRiverwolford]

@@katech6020 I knew he programmed these demonstrations, but I didn't know exactly what he used to do so, so thanks for that! I'll have to check it out at some point.

[@krishna25MO]

Even if I don't understand all technical aspects of your videos I really appreciate the visualization that give me an deeper understanding of mathematical problems. Thanks!

[@lindsay3917]

This was awesome! Have you considered doing a follow-up on Dirichlet's theorem about Chebyshev's bias? For example, when you showed the histogram of primes 1, 3, 7, and 9 mod 10, there is a bias towards 3 and 7 mod 10 (because these are non-squares). Even though the categories all have 25% in the limit, there is quantitatively more primes 3 and 7 mod 10. The primes race is really compelling and not too hard to understand.

[@2ndOfficerCHL]

"Euler's totient function." I swear, Euler had a hand in everything.

[@gizmodobaggins7040]

Euler, the madlad of math.

[@jdawggghg]

euler’s hoarding problem

[@Crumbling_Vortex]

Soo... you're telling me I need to look up Euler

[@BobStBubba]

Euler was overcompensating for for the non-phonetic pronunciation of the spelling of his name. First class, every semester of math, teacher calls his name, "Leonard Yuler." "That's pronounced LeonHard Oiler." "Well, it looks like Yuler to me." "I can't help what it looks like to you...." and over time, decides to write 800-plus mathematical treatises just to make math teachers' lives everywhere miserable. And also, ours.

[@BobStBubba]

"Euler's totient function" sounds like something discovered not by Euler himself, but by his mother, during his toilet training, during the years he was studying enuresis. "He's got to get control of his totient function, or he'll never leave home"

[@sirsholar]

When you discover math before you learn the math theorem, then the theorem becomes your friend instead of an arbitrary inconvenience.

[@minebloxgx1780]

Well said

[@n_x1891]

You're a fucking genius.

[@BladeOfLight16]

I think this is an expression of the fact that math is typically motivated by the goal of explaining some particular phenomenon. For the Greeks, the were trying to explain and model the properties of geometry. For Newton, he was trying to explain motion. For Einstein, he was trying to explain weird things about gravity (although he took the math of general relativity from others). As math has grown increasingly complex over the centuries, it developed its own, non-physical phenomenons of interest. It's this sense of discovering patterns and relationships and being able to describe and explain them relatively simply that motivates us as humans to do math, and playing with problems on your own leads you to that sense in a way that memorizing and practicing a set of theorems can't.

[@heroricspiritfreinen38]

@@BladeOfLight16 waffle

[@yuvanmarimuthu4909]

@@heroricspiritfreinen38 not really

[@scottleung9587]

This was beautiful to watch - and as a Math major, I learned more than you could possibly imagine. Thanks a million!

[@martins.1584]

This is where it started for me. A recommendation of this video is how I found your channel and it let to sth important, at least for me. I am going to start teaching theoretical computer science soon and, although it is not really the topic of your channel, I will try to use as many of your tips on conveying ideas visually as I can. Thanks and keep up the great work!

[@TraceyIsNotMaryGrace]

Awesome !!

[@PiercingSight]

"If you effectively reinvent ... before you've seen it defined... then when you do learn those topics, you'll see them as familiar friends, not as arbitrary definitions." This is my favorite thing about messing around with math and numbers, finding patterns, testing different ways of measuring their properties and more. I didn't know what integrals were before high school, but I knew that if I added up all the space underneath a graphed line or curve, then that would be useful for say... adding up the total distance a car travels while only knowing it's speed over time. When I finally learned about integrals, it made the topic so much more exciting for me. Thank you for continuing to make math fun and interesting for everyone who sees your videos!

[@jasonbellmusic3091]

Same... That really hit me right in the face when I heard it. I'd been looking at a number pattern thingy (the description isn't clear whenever I try to explain it so feel free to skip to the next paragraph) where I try to see how soon a digit repeats itself when raising a number to an integer power which I increase, and I found various patterns which seemed almost arbitrary. In the end, I spoke about it with my brother, and he told me how it was related to this very totient function, and gave me a brief explanation. So once I saw it even in this video, I felt more familiar and certain with myself.

[@WobblycogsUk]

As someone who understand only a little maths it's very easy to see a diagram like that and think there's some deeper truth to it. The way you explained that there isn't was absolutely brilliant, thanks.

[@sunnylilacs]

Wobblycogs Workshop Why does learning the explanation behind it make you think there isn’t deeper truth?

[@wasd____]

@@sunnylilacs Same reason nobody believes in unicorns - they don't _need_ to exist because there is no evidence requiring unicorns as an explanation.

[@dopaminecloud]

@@sunnylilacs it's very easy to get swept up in patterns and start making broken logic leaps, consequence of the brain liking them so much there's an entire mental condition based on an extreme version of this tendency to get stuck to patterns

[@hermanubis96]

Dopamine Cloud so why are there patterns at all?

[@wasd____]

@@hermanubis96 Because pattern recognition was adaptive and beneficial, therefore it was selected for during human evolution.

[@Kloiyd]

The video explained amazingly how this spiral came to be and made me understand a concept I previously thought I wouldn’t be able to understand.

[@thebrownmalcolm9498]

A buddy sent this to me a few minutes ago. This is fantastic. Liked and subscribed. I’m glad something like this has 3.5 million views.

[@valasfar1557]

The number theory ones are always so interesting!

[@erikkonstas]

I feel like number theory is more useless than other branches but it poses some interesting and often difficult problems.

[@mironhunia300]

@@erikkonstas Number theory is the basis for cryptography, so it's pretty much one of the most useful branches of mathematics right now.

[@erikkonstas]

@@mironhunia300 Although it compares less in usefulness to e.g. calculus. I agree that every branch of mathematics which potentially has an application is very useful, I'm just doing a comparison. Personal opinions might differ, but eh.

[@osolomons]

All the ones are always so interesting!

[@99bits46]

number theory is bs

[@BorniWolf]

Hi, I'm a mathematician, and have to say, WOW, I enjoy your videos a lot, have just recommended your channel to a friend of mine who teaches in high school to show your vids to his students, perhaps, with your help, more young talented students will be "lured" to study mathematics:) thank you very much for your work!!

[@RipRoaringGarage]

What field are you? I was doing Representation Theory and Number Theory, with a dash of Hyperbolic Geom....I wish I could get back to that. Its just that there is no way for me to return...and when youre sitting on an important proof, it is maddening.

[@trixylizard6970]

I'm 43 and it made me pick up the books!

[@SupaJay2]

True! Also I wonder if perfect numbers would do something...

[@opticandersonopticanderson3364]

​@@RipRoaringGarage😂 anyone can claim to be a mathetician online.

[@19Szabolcs91]

This is amazing, all of this. From the original spiral shape to the explanation to the conclusion about learning at the end.

[@debojitsikdar2046]

This is pure Mathematical bliss. Thank you so much for this, as you said, now I will be more familiar with these concepts when I go into them more deeply!

[@ano2math5]

Oh, truly a piece of art. I’ve never seen a movie which expresses the cliché that “math is beautiful” better than this video!!! I love this!

[@ianwalker6546]

I love your final point. I remember when I was at school, adding up the number of spots on a normal six-sided die, and then independently asking myself, and coming up with, the formula for "how many spots on a die on any number of sides?" - a question that was probably helped due to my D&D hobby making me familiar with the idea of dice with different numbers of sides." So I independently "invented" the formula for the triangle numbers, which is not a particularly great mathematical feat, but did allow me to stun a teacher who set the classic "Add up the numbers from 1 to 100" by answering it within a few seconds. Great video!

[@jakefromstatefarm6969]

You gaussed them!

[@mcmonkey26]

same

[@yuraje4k348]

i loved the first point too. "How pretty but pointless patterns in polar plots of primes prompt pretty important ponderings on properties of those primes"

[@Adventurin_hobbit]

Yeah exactly 💯

[@yuraje4k348]

@@Adventurin_hobbit yo sir give more formulaes

[@Oblivionator100]

I really like how this visualization shows the principles of emergent properties. Given a set of rules, any system, complex or simple, will have properties emerge that are non-obvious from the inception of the system. This is one of my favorite observations of universal principles.

[@SporkleBM]

The last bit? about these topics coming back as familiar subjects instead of arbitrary definitions, really solidified my love of youtube education. for channels especially like this, minutephysics, etc

[@rianby64]

And I almost cried after reading "Be playful". Really amazing conclusions you gave us here!

[@Damathematician]

I feel yah, me too man.

[@rianby64]

kzbin.info/www/bejne/n3fYg2xtjsiGgq8 - slightly similar concept, but not. Conclusions are totally different.

[@shrimpchem]

This is analogous to showing a meme to your parent and instead of saying “oh cool”, they give you a piece of life advice

[@DillsArtThing]

._.

[@johnnyswatts]

It's more like showing a meme to your parents and they say "oh, cool" and then share a really deep story from their lives that relates to that meme, showing you worlds beyond and making you feel really good and loved.

[@NortheastGamer]

@@johnnyswatts There are two types of people, those who listened to their parents' stories and those who rolled their eyes.

[@shatterdpixel]

NortheastGamer Or a mystical third kind where their parents just yelled at them

[@Jonesybabie]

Shatterdpixel And a random fourth whose parents didn't say anything at all... But the children still heard everything that needed to be said and eventually learned why primes form spirals. Clearly it's used in the flex capacitor to initialize time travel 🤓

[@dawislv]

This is simply amazing video and clarity level. Personally for me this way of thinking is a striking resemblance of "when you eliminate whatever is impossible, whatever remains, however incredible, must be true" and reminded me of very old days when I was thinking about way of defining set all primes myself which lead me to a c++ program which ( how I later learned) was a bad implementation of Sieve of Eratosthenes. The formula for distribution 1/Fi(N) makes total sense when explained so clearly like that.

[@derekmz]

The way you seamlessly explained the jargon for modulo was perfect

[@potatok123]

*The ultimate connect the dots game*

[@flanbenflen9069]

300 likes comment-less?? Don't think so.

[@AnarchoAmericium]

*takes a deep breath* Let me take the time to talk to you about category theory.

[@josephdestaubin7426]

Indeed, and on so many levels. ☺

[@potato457]

IMPOSTER

[@potato457]

I AM THE TRUE POTATO

[@jaredgarbo3679]

"3 is slightly less than Pi" You have angered the engineers.

[@gabor6259]

"Pi is 1." /a physicist/

[@SuperPol1981]

Engineers would be the first to simplify pi to 3. You're thinking about mathematicians. Or school teachers or lawyers.

[@bledlbledlbledl]

The engineers aren't the ones you have to watch out for on this one. It's the slapstick comedians. (lemon-meringue pi)

[@tdiaz5555]

@@SuperPol1981 The (running) joke is that an engineer would say that pi = 3, while the statement here is pi > 3.

[@17lvlham]

Even during desinging of simple DSP for my radio amateur transceiver, I've been taking pi to about 9 decimal places to have enough frequency accuracy (nearly 10 Hz) in my "narrow" working band (30 MHz). And I wouldn't be surprised, if serious engineers take pi much more accurate. E.g. in automatic control theory, while estimaing safety margin of some closed-loop control system.

[@TheDentrassi]

This is great. I've stumbled into maths from archaeology and art. Wanting to reproduce various neolithic and la tene designs and monuments on paper and mucking around with a compass and rule. I've found learning mathematics from this visual, geometric perspective has made a lot of things click in a way it wouldn't before. I really appreciate seeing mathematical concepts visualised like this.

[@kiiometric]

This was so entertaining I didn't even realize that was 22 minutes long, I love this♥️

[@fnalley1761]

I didn't look at the time of the video before starting, and kind of assumed it was about 10 minutes. Then at the end, I thought...wow, that must have been only 5 minutes. LOL

[@isabellaegan5051]

Same, you were the one that made me look at the time stamp for the first time

[@arnbrandy]

"How pretty but pointless patterns on polar plots of primes prompt pretty important ponderings on properties of those primes." C'mon man, let me just watch a minute or two of the video before forcing me to like it.

[@vieuxnez]

P'shaw.

[@echozero8213]

Plliteration

[@Jlouise95]

important-->purposeful

[@funkysagancat3295]

Also the patterns cannot be pointless when they are clearly coming from a bunch of points

[@PandemoniumMeltDown]

There is beauty in innocence

[@Shuizid]

Great video! Also great way of showing how things look in scales when zooming out and one set of spirals transitions into another one with the implication that we could zoom more and more only to find now sets of more and more spirals.

[@r50142]

I love the little PI buddies you're making. Honestly keeps me motivated to keep coding.

[@syedarslanalishah6905]

I'm smiling before even the video is started :)

[@blackcat5771]

Same :)

[@anandsuralkar2947]

Samea

[@heest2876]

First you click on the video, then thumbsup, then it loads :D

[@see_conspiracy_inevery_tri6991]

Why? Something is wrong with you guys? Maybe you need professional help?

[@The_Rising_Dragon]

(.❛ ᴗ ❛.)

[@smallbar2012]

When I was in sixth grade, I realized that the difference between any two consecutive squares was equal to the sum of their square roots. I was blown away by this fact, presented it to my teacher, and was ecstatic to learn that that tidbit generalized to the Difference of Two Squares. I then spent the next three years telling people I had discovered a theorem on my own, and I was so proud of what I had discovered by playing around and chasing patterns.

[@elkraftaren245]

This is the exact same thing I discovered, litterally exact same story. mind blowing

[@seesaw41]

In 6th grade?

[@smallbar2012]

Yep! I got bored a lot in school. Haha.

[@GrandAdmiralMitthrawnuruodo]

Thank you so much for that video! You once again showed how beautiful mathematics is and how it can help us understand the world! Your videos always give me chills. But the good ones of course.

[@therealzilch]

Simply wow. To my credit, I did see how the patterns must have something to do with how polar coordinates work. But this went way beyond that. Thanks, from someone who plays around with prime and coprime polyrhythms and polymeters, Scott

[@luistorh]

The last minute of your talk was profound, enlightening and valuable: the connections of deep math concepts to many manifestations of reality. Thanks.

[@douglasthompson9070]

Humans love to find patterns so they can figure out why a pattern exists.

[@smilelikeUmeanit90]

Patterns love to find humans. Oops.

[@glaswasser]

do we find patterns beautiful because everything is in a pattern - or do we find patterns beautiful because we were "programmed" to like patterns, or both?

[@hanguyenthu9691]

@@glaswasser Now silly as this question/joke might seem, the answer is quite worth it to look into. You see, pattern give us an important ability: to predict. Then of course, creatures that are programed to see patterns might predict things better, and be better at living at a whole. And what is a better reason to look for patterns, than its beauty?

[@whitechocolate4384]

Patterns exist. Humans are keen towards them because our brains allow us to recognize them. Patterns are caused by stimuli. We are intelligent enough to domesticate those stimuli if we comprehend them.

[@venkatbabu186]

Patterns are sequence to follow for direction and routing and assessment of speed and vectors.

[@gaanasonata6582]

Hi 3Blue1Brown, I love your content... as a high school student and ‘mathlete’ I was extremely excited when watching this video! You mentioned that the proof of Dirichlet’s was quite complex... can you explain it on your channel? Your visual style of explanation would be amazing to learn that!

[@davidpederson2905]

This video has had me thinking about this for a long, long time. I love how it is so easy for the mind to "see" the rays. But the "spirals" are just discontinuous dots that happen to be near each other, and the human brain is compelled to connect the dots. I wonder how often we get fooled by things like sub atomic particle data or astrophysics data that is really discontinuous, and assign some kind of relation that might not really be there, but "kind of" explains the data so we stop looking for other explanations. Fine job on the whole idea, and the presentation. Many thanks.

[@worzo1284]

Aside from the astonishingly clear explanation of this problem, this is a great insight into why many people find maths difficult. Effective learning is about making connections between things. We often teach maths as "learn this set of rules", which has very few connections. Exploring patterns and then explaining them as this video does is much more powerful.

[@foooooooont4679]

"Why do prime numbers make these spirals?" me before the video: how tf should i know that me after the video: *what are prime numbers*

[@FeedEgg]

They are portals into and out of our minds simultaneously....yea pretty nuts i know.

[@foooooooont4679]

@@FeedEgg god, is that you?

[@FeedEgg]

@@foooooooont4679 one of them...shh

[@foooooooont4679]

@@FeedEgg ok i will keep my mouth shut

[@FeedEgg]

@@foooooooont4679 lol do not be afraid, they exist just not in the way you think, agnostic, all i know is that i know nothing at all.

[@madsjakobsen9824]

I am writing a big school paper on RSA encryption, and wanted to watch some 3b1b videos so i just took one i hadnt watched. And boy does it feel satisfying when you started talking about eulers totient and coprime numbers because i have been learning so much about that stuff. Great video

[@luminous2585]

I remember learning about reside classes in school. We had this little experiment going on where the computer science and math teachers tag teamed us for a couple lessons to teach us about RSA. Honestly, that was a cool idea and I wish more students got to experience something like it. Seeing how different subjects connect with each other is really special.

[@LoUiSvsMiKu]

1am: i have to sleep *3b1b uploads*

[@green0563]

Same.

[@sundaralingams8083]

Exactly same. 😂

[@alphaexpress6881]

Same.

[@AnkitSingh-wq2rk]

For me currently 12 am but same

[@ishaanparikh485]

Same

[@shahirkazi8766]

This video made me... feel emotions that I can't quite put into words.

[@programaths]

Forty Two.

[@romanpfarrhofer]

try to put it in numbers instead :)

[@electronicmusicwave7459]

i know what u mean. me too...

[@chasebender7473]

@seba cea Andre Weil once said that understanding a problem that you have been working on endlessly can lead to a feeling of ecstasy for weeks at a time

[@Damathematician]

The end of this video took me to such an emotional high. Its nice to see others who care so deeply about a subject you also care so deeply about. In a way it was like our spirits became one ... although philosophically I am not sold on 'spirits', that's the language I have to use to describe this feeling.

[@SabrinaXe]

The end was so cute and touching. I love your bravery for not getting too caught up with arbitrary patterns and seeing beyond them

[@linklegends22]

This is such a beautifully clear video. I've seen this prime spiral meme before and like you said thought it was due to some mysterious property of primes. Thank you for demystifying this and somehow leaving me even more amazed by the simplicity of the mathematics causing it and the more interesting topic that it brushes up against.

[@gregmartin6341]

Note that you might be thinking of the "Ulam spiral", which is a different spiral- and prime-related picture...!

[@Ironpecker]

I cant wrap around how math can be so beatiful, it's like reading a really good novel that has many intresting characters and plots that are always more deep and connected that they lead you on at the start. Sometimes it requires more work to piece all the parts together but man the result is incredible

[@jamesr2936]

Nice analogy well said! And in the case of our universe, math is the language in which the novel is written. As Kepler said :)

[@kevinpruett6424]

​​@@jamesr2936it's not as fancy as an entire novel filled with gestures... It's more like musiComposition. (It repeatStatic loops when charted, but cannot "break the mold" via willpower. It is the environmental medium, the reflecTensor allowing language). Language, on the other hand, is not exact or predictable, for having synonym varianTone stretching.

[@moothecow6908]

Ok i relate so much to the end of this video because im in high school and i just learned derivatives this year but ive been doing simple derivatives forever. I thought the fact that there was a specific ratio of the slopes of like whole number values of x^2 was really interesting and that each had a specific relationship to the previous and only now do i realize that that was derivatives the whole time and its an amazing feeling

[@user-nl2kr1nk9s]

I remember asking my high school teacher when I would use maths in life, his answer was an uninspiring "probably never". The correct answer would have been any time you want to understand anything.

[@kristypolymath1359]

You use math involuntarily. Just by walking in such a manner that you don't bump in anything, or that you are able to tie good knots in your shoes, requires you to use math. You just don't think of them in conventional manner. You're still making calculations.

[@violinscratcher]

Kristy Whalen And while hearing music: You do differential calculus to „translate“ frequencies into pitch and you cancel fractions when you hear intervalls and/or feel their tension.

[@jonadabtheunsightly]

Yeah, but for most of his students, that's the same answer. 99% of the population goes through life actively avoiding understanding anything ever. They get through school by memorizing so they don't have to understand, and then they spend their entire lives just doing the same stuff over and over by rote. Idealistic young math teachers fresh out of college often don't know this, but by the time they've been teaching for 30 years, they're usually a bit more jaded. You can tell which kind of student is which in geometry class, when you assign proofs. Most of the students will (at best) come up with the official stock proofs, which are usually 3-5 steps long, either because they copy off someone else, or because they dutifully memorize every single theorem, including all the ones with 3 step proofs. High school geometry texts are designed so that if you do this, you never need to put together a proof more than about 5 steps long, and also so that all the theorems you need for each of your proofs are within the last chapter or two. When you see a student who doesn't bother to memorize all the trivial and obvious theorems, so then his proofs are 30+ steps long but entirely valid, you know you're dealing with someone who actually understands what's going on. He can remember the important theorems from ten chapters back, because he knows what they mean; and the trivial theorems he can derive on the spot as needed, because they're trivial. You'll typically have about one such student per year, assuming you're teaching 5 classes of 20 students, give or take.

[@user-nl2kr1nk9s]

@@jonadabtheunsightly Sad but true.

[@moodberry]

@@jonadabtheunsightly I guess you are or were a teacher, right? So, I also wonder what the proportion, as a percentage, is of students who "get it" and those who don't? I would love to see a plot of these numbers instead of a prime plot and see if patterns emerge. If so, then I think we could predict whether society is getting smarter, declining, or staying constant. Hmmm?

[@GregL-zt4xf]

You, sir, are one of the most effective teachers on the planet.

[@wolfelkan8183]

One implication he didn't go into: When plotting the numbers in whole number radians, each new number was 1/2pi rotations from the last one. So, the numbers made a spiral arm every time they encountered a number that was close to the denominator of a rational approximation of 1/2pi (that is to say, close to twice the numerator of a rational approximation of pi itself). But what if we didn't want to make spirals? What if we wanted all of our points to be as far away from other points as possible, *in every direction*? (Why we would need to do this is a point I'll come back to later.) If you're making spiral arms, there's a lot of space in between the arms that's wasted, and much less space between two neighbors on the same arm. Is there a way to avoid this? Well, if we want to find a number that gives us no spirals, we need it to have as few rational approximations as possible, (some of you might see where I'm going with this) we can look at continued fractions, since as explained in that Mathologer video, every time you encounter a large number in a number's continued fraction, you can truncate the sequence there and get a pretty good approximation. Thus, the ultimate not-close-to-any-rational-number number would have a continued fraction with numbers as low as possible. Ideally, made up of all 1's. This number happens to be (sqrt(5)+1)/2, known as the Golden Ratio. But getting back to why we would need to find points as far away from each other as possible: Well, what if we were a plant putting out seeds? We have a chemical process that rotates by a certain amount and then makes a seed. And we want those seeds to be spread out as efficiently as possible so that they don't have to compete for resources. If you've heard that the Golden Ratio shows up in nature, this is why.

[@nichtrichtigrum]

I really appreciate your comment pointing towards the connections between math and nature and I think it would make another great video (I hope @3blue1brown reads this)! Do you maybe have a source for this that I can go to?

[@legacykevin]

Awesome!

[@JordanMetroidManiac]

Yes, people should give the golden ratio more attention! It's got some crazy (cool) things too! Try this out: Find the line that connects the two inflection points of a quartic polynomial curve. Then, measure the distance between the outer intersections (the rightmost point and the leftmost point) and the inner intersections (the inflection points). It turns out that, provided that four distinct intersections exist, the ratio of the inner segment (the distance between the inflection points) to the outer segments (the distance between each of the outer intersections and the nearest inner intersection) is exactly the golden ratio. Furthermore, the two smaller areas enclosed (on the left and right) by the inflection line and the quartic curve are each exactly half the size of the larger area (in the middle). Why this happens probably comes down to a nasty algebraic nightmare with calculus, and things simplify to the golden ratio and whatnot. I'm sure it's possible to prove it. I tried to do it myself but got lost in the awfully complicated algebra (trust me, it's ridiculous). Maybe there's a neater and more elegant proof than that, though. 3Blue1Brown? Care to tackle this one?

[@VishalSingh-jn6qw]

Pheww!!!!! So long thst i couldn't help liking!!

[@Kashish290695]

m.kzbin.info/www/bejne/qZubhJpuptOdhck

[@Paroxysm80]

I just randomly stumbled upon this, and it has me absolutely fascinated (from both the resulting math and the lucidity of the video/explanation itself). Amongst my other playlists for memes, drumming, etc., I now have one titled "Beautiful Math". I feel compelled to fill it with others and take the time to understand it all! Thank you so much for creating this incredible lesson! :)

[@abuzabid6473]

This shows even simple things in mathematics can lead to something stunningly beautiful things, this just amazes me more that how beautiful mathematics is!

[@forrest11]

This channel is absolutely phenomenal, you deserve much more viewers

[@FacultyofKhan]

Oh yes, waited such a long time for this! Quick Request: since you're doing Number Theory, can you prove Fermat's Last Theorem? I believe the proof is quite trivial, so it shouldn't be too bad :P

[@runningcrocodile8051]

Yeah, Fermat's last theorem is an easy one.. definitely should be a video. In fact, I just found a nice proof for it, but I'm afraid it won't fit in this youtube comment.

[@chumbucket6989]

He's addressed this: www.reddit.com/r/3Blue1Brown/comments/7aubxv/fermats_last_theorem/

[@FacultyofKhan]

@@chumbucket6989 Aww nooo, is the project too big to fit on the margin of his paper?

[@nchoosekmath]

@@runningcrocodile8051 lol nice one.

[@chumbucket6989]

@@FacultyofKhan This is what he said: "I'm not saying no, but let's just say this would be a very big project :) Certainly some special cases might be doable and interesting."

[@pikasfed]

2:25 that animation and change of music was utterly beautiful, that type of beauty you wouldn't expect to find, yet still it's there, waiting to be discovered.

[@roguelegend4945]

thank you ' for all the videos' i love to watch them and find them very interesting, it brightens my days...

[@AdrianHereToHelp]

My mind is blown. What a phenomenal and beautiful video; thank you for making this.

[@MindMathMoney]

🕊️ *The beauty of mathematics only shows itself to more patient followers.* 🕊️

[@jaredt5127]

Math teachers be takin notes on this channel. Superb

[@ancbi]

Or to those follower of 3B1B even if they are less patient.

[@STAWBsOrio]

Absolutely stunning. I am a part-time mathematics teacher myself and the epilogue was truly inspirational. Thank you.

[@RaoufAthar]

This is a wonderful video. The amount of effort gone into making the video and the knowledge are praiseworthy.

[@Bbb78651]

"So be playful!" Brilliant words, and brilliant, brilliant video. Thank you Grant.

[@Adam-jo3tr]

I love the way you take the time to teach math jargon and other tidbits in these videos. So well done. I wish every single lecture was like this

[@AapoJoki]

This video in a nutshell: "That was a pretty dumb question, but here's a _really_ good answer to it"

[@brokkrep]

I also thought why to ask that, because this graph is completely man-made so it is no wonder such thing happens.

[@Fralexion]

"...and that retroactively means it _wasn't_ dumb, because curiosity lead to learning something"

[@portaadonai]

Design: ...>>>oooOOOooo

[@Fralexion]

​@@portaadonai Your reply has nothing to do with the comments above it, and is very clearly an attempt to derail the conversation into a tiresome debate about intelligent design theory. Please put your digression somewhere else.

[@callahans44]

@@portaadonai I'm pretty sure one gets a straight line with nos, but he got a spiral using pi and radians such as 2 pi. Of course you a spiral no matter what unless you get a near circle. So no randomness. It's how these guys saw patterns within them w/o drugs is the lesson here. I think.

[@void-qh8uc]

I love this channel so much! Using my free time to study mathematics (also other sciences) is awesome :)

[@tomcox1983]

A beautiful production with a beautiful message. Thank you!

[@TheAaroneusthegreat]

When you zoomed out from the initial set of prime numbers I got chills all over seeing that beautiful spiral come out of the numbers like that. Bravo! This is fascinating!

[@dumdumdumdum8804]

But the spiral is coming from the whole numbers also.

[@ayushbhardwas]

When mathematicians get inspired by chemistry, remainders become residues.

[@Allangulon]

They're looking for a Solution!

[@ayushbhardwas]

@@Allangulon 😂😂😂

[@jagtan13]

@@Allangulon hey carefull you wouldn't want a Suspension!

[@MahendraSingh-nb7ui]

Haha 🤣

[@Anvilshock]

@@jagtan13 Suspensions are pure physics, though. They work without chemistry, mind.

[@MrHailstorm00]

Whenever I feel discouraged by humanity, I come to this channel and get courage from knowing this video still can amass millions of views

[@AlecAkin]

The “residue class mod 6” made a whole lot more sense when I equated it to the modulus operator often used in programming - especially with the Fizz Buzz problem. Now I want to see Fizz Buzz on a radial graph!

[@eamonnsiocain6454]

Precisely! Discovering patterns on one's own greatly enriches one's understanding of Maths and one's appreciation for its intrinsic beauty.

[@Traagst]

That spiraled out of control quickly..

[@mikedamacenos]

Underated comment

[@davidherz9968]

@@mikedamacenos why is it out of control? Since when has infinity been out of control? Just out of reach, out of sight, out of mind.

[@epaulander2268]

This is amazing. It's so deep and so simple at the same time.

[@rockysmith6105]

That intro was already spectacular, that alitteration was immense

[@alapanbera8259]

Seriouly , i never realized there is so much beauty hidden in math before watching your videos..thank you 3blue1brown❤️❤️❤️

[@Cardgames4children]

Math has a lot of subtle patterns, often too convoluted to see the whole picture and beauty all at once. But with each careful step, you can get closer to seeing how various things and ideas/concepts fit together, and that, in the end, can give you a deep appreciation for how it all works. It's really cool just how abundant patterns can seem around us.

[@yovliporat8608]

I literally finished a number theory course in my degree two weeks ago, and was tested on almost everything you brought up in the video!

[@RO-pp4kx]

this video is incredibly informative and well made. Thank you!

[@wailingalen]

I am not a mathematical phenom or engineer, but ai do find immense beauty in the visualizations of mathematical concepts! Like this, visualizing Fourier transform, visualizing how to turn a circle inside out, mathematical proofs, etc!

[@billymcnutt116]

I commend all the mathematicians who made these discoveries before computers were invented. 👏👏

[@carlquitter4987]

Hey I’m going through a very tough and stressful times and I wanted to say that seeing your video in my feed just made me smile and actually really excited me. Thank you

[@hugoehhh]

Hope the times are doing you better my friend

[@fazelamirvahedi9911]

Thank you a million. Watching mere mathematics videos like this always makes me thrilled. I was wondering since we used the polar coordinates in this video, we already expected to see the approximations like 22/7 ~= pi, or other ones, so I was curious to see where these approximations come from? Where is the origin of the pi number and why is it irrational?