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

I feel this comment is underrated.

@@juhonuorala3512 That's a nice line.

lmfao

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

you now have 0 likes in 8 bit systems.

Yep!!!

i was just about to say beautiful

Couldn't agree more!!!

also Amazingk!!

How about bootyful?

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).

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.

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

@@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.

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

That's actually really cool

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

977-311=666

@@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.

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

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

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 KRplus, but I really appreciate what Grant does.

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

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

totally agree... subscribed right a way

@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

@@SARGAMESH The reason why the video “glitches” when he zooms is that the video has a certain maximum bitrate. KRplus’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.

Arpan Dhatt mkbhd proved it with his 1000 upload test

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

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.

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

it loses so much after first viewing but is still brilliant

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

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

@@TheHwiwonKim do they turen abck into beams?

Time Stamp?

LMFAO

LMAO
RIP.

@@hybmnzz2658 heroin or hero in? x)

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

@@aeiouaeiouaeiou clever junkie

@@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.

@@_stockfootage Depends on the country.

@@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.

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

Keep going!! Math is pretty cool.

who would

@@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.

@@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.

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

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

@@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.

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

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

@@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.

Awesome !!

Euler, the madlad of math.

euler’s hoarding problem

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

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.

"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"

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.

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. ;)

@@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.

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

"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.

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

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

@@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.

All the ones are always so interesting!

number theory is bs

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

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

@@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

Dopamine Cloud so why are there patterns at all?

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

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.

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

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.

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!

@@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.)

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

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

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

Indeed, and on so many levels. ☺

IMPOSTER

I AM THE TRUE POTATO

Well said

You're a fucking genius.

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.

@@BladeOfLight16 waffle

@@heroricspiritfreinen38 not really

I feel yah, me too man.

krplus.net/bidio/maqthGlpi5q3fX4 - slightly similar concept, but not. Conclusions are totally different.

Same :)

Samea

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

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

(.❛ ᴗ ❛.)

You gaussed them!

same

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"

Yeah exactly 💯

@@Adventurin_hobbit yo sir give more formulaes

._.

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.

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

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

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 🤓

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.

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

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

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

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

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

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

In 6th grade?

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

Patterns love to find humans. Oops.

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?

@@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?

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.

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

"Pi is 1." /a physicist/

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

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

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

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.

Same.

Exactly same. 😂

Same.

For me currently 12 am but same

Same

P'shaw.

Plliteration

important-->purposeful

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

There is beauty in innocence

Forty Two.

try to put it in numbers instead :)

i know what u mean. me too...

@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

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.

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.

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

@@FeedEgg god, is that you?

@@foooooooont4679 one of them...shh

@@FeedEgg ok i will keep my mouth shut

@@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.

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

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.

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

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

@@runningcrocodile8051 lol nice one.

@@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."

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

​​@@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.

They're looking for a Solution!

@@Allangulon 😂😂😂

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

Haha 🤣

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

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

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

Design: ...>>>oooOOOooo

​@@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.

@@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.

I Watched this at exactly 02:00!(24hr clock)

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?

Awesome!

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?

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

m.krplus.net/bidio/o85whZdqo6XOgJg

TUSHAR SHAILY as far as I can see, all schools everywhere rush to the ‘answer’ when really it is the question that is really interesting.

I'm thankful to be taught this now. I feel for all the people now gone (or still alive but nevertheless will never have the opportunity) who may have been completely enamored by this privilege. But yes, I do wish this was available during school. No doubt there are some lucky students out there with splendid teachers at this moment

Schools are forced to ensure kids can pass tests more than anything else. In the UK the benchmark for GCSE exams (sat when you're 15/16) is (well, was) a C grade. Getting a D-grade kid to a C meant a lot to the school, so way more effort was expended by teachers in that area. The high flying kids, those that could get an A without too much trouble, weren't pushed anywhere near as much. Not the fault of the teachers, I might add. The school doesn't care if an A* possible kid only gets an A, that won't really affect stuff like funding. It's all about getting kids to pass. I'm not saying the struggling kids shouldn't be helped, but it shouldn't be that schools have to prioritise them any more than kids with a high potential in that subject.
They recently changed the grading from letters (A* to F) to numbers (9 to 1). Why? No doubt there are 'reasons' but it does not seem a priority to me. But that shows the nature of schools nowadays.

I was lucky enough to have a math teacher who showed us the beauty of math (in the 80’s, no youtube just chalk but he did it). This guy even gave up his pauses if anyone didn’t grasp anything and would explain again or explain more (out of scope for the exams) for those who were interested. My wife didn’t have this luck and always thought she was bad at everything math until I started explaining things I remembered after we got married. Long story short, she went back to university while working full time (didn’t attend most classes), did every year in half the time and has since become top of the field in her profession and gives guest colleges at several universities. She’s the perfect example of how the spark was not lit up because her teachers failed. This is why youtube is such an important tool right now where capable and driven people can enlighten the people who are interested and hopefully light many sparks!

I think i may have learned more maths from youtubers at this point then i did in college. and i minored in mathematics.

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.

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.

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.

@@jonadabtheunsightly Sad but true.

@@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?

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.

Underated comment

@@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.

Wild Abra used teleport.

…4 4 used TM55!

Poor computer

A wild Abra appeared
Go, 1
Wild Abra used 1 is not a prime number
It's super effective
1 commited suicide
Go, 0
Wild Abra used Teleport
Wild Abra fled

Hope the times are doing you better my friend