Why do prime numbers make these spirals?

  वेळा पाहिला 2,766,141




A story of mathematical play.
Home page: www.3blue1brown.com
Brought to you by you: 3b1b.co/spiral-thanks
Based on this Math Stack Exchange post:
Want to learn more about rational approximations? See this Mathologer video.
Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video:
Dirichlet's paper:
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.
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.
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:
Stream the music on Spotify:
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.
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Marah Wolsfeld
Marah Wolsfeld 11 तासांपूर्वी
So, if someone plans to animate hyperspace, they can just take prime numbers!
ayush anand
ayush anand 14 तासांपूर्वी
Beautiful 👍
David Sysko
David Sysko दिवसापूर्वी
Beautifully visualized, clearly explained. Bonus: the plug for advanced education toward the end.
Letys 2 दिवसांपूर्वी
Christopher Maupin
Christopher Maupin 2 दिवसांपूर्वी
The fact that 2.7 million people watched this gives me some hope for mankind.
Duesternis 2 दिवसांपूर्वी
Easy Fix: Write the numbers in a line!
Sun Daojun
Sun Daojun 2 दिवसांपूर्वी
Those spirals are fascinating and the explanations are amazing.
James the Bear
James the Bear 2 दिवसांपूर्वी
Everything is so much simpler when you understand that everything that exists was created and intended and dine with purpose by an infinitely intelligent creator. It's our privilege and joy to find the answers to the mysteries He gave us.
Michael Kilgore
Michael Kilgore 2 दिवसांपूर्वी
Dave Rubin. You sound like Dave Rubin. I've been trying to place it for weeks lol
Julie Johnson
Julie Johnson 2 दिवसांपूर्वी
The Creator(of the universe) is an organised artist
Jari Sipilainen
Jari Sipilainen 3 दिवसांपूर्वी
21:28 yes yes numbers are fun. its kinda correct and then pyramid measures was kinda planned i guess LOL everything related close enought. it all looks like planned LOL but what if not. will it mean end of life? if you layer that on US map what you see. was it planned?
Jari Sipilainen
Jari Sipilainen 3 दिवसांपूर्वी
6:15 thank alot NOT 1st and last are not lottery numbers but all rest are and do not never use them xD they not win, but diaconal will DID!!!
Shreya Gokhale
Shreya Gokhale 3 दिवसांपूर्वी
Dude, you make me love Maths more and more every day
joel ivan cittar
joel ivan cittar 3 दिवसांपूर्वी
The secrets of the universe are encoded in the first numbers. You just have to know how to read them, to obtain all the universal information available.
Jeff Murphy
Jeff Murphy 3 दिवसांपूर्वी
I apologise for giving this a thumbs down, but there were 946 dislikes at the time of this comment, and I was not I the mood to see a non-prime number under this video.
Bruno 4 दिवसांपूर्वी
hey blue did you ever attempt solving one of the millennium problems? this looks like a clue to the Riemann problem for any mathematician out there you can win 1mil$
Nuby 4 दिवसांपूर्वी
You can make a religion out of this.
Hunter Watkins
Hunter Watkins 4 दिवसांपूर्वी
The unusual kendo ordinarily bang because ray bailly whisper during a irate baboon. terrific, bloody parallelogram
Ellie Mäkelä
Ellie Mäkelä 4 दिवसांपूर्वी
math hasn't made me this happy since 5th grade
Dip Bhowmik
Dip Bhowmik 4 दिवसांपूर्वी
Everyone who came here in this video are knowledge seekers.
CANNAHEALTH 4 दिवसांपूर्वी
thomas dunham
thomas dunham 5 दिवसांपूर्वी
i,m 844 mos. old ,4 goes into ,=211 see how it all works out.....hmmmm
Alex Hando
Alex Hando 5 दिवसांपूर्वी
13:52 Oh gosh I think I just got hallucinations from that
Alan Cliff
Alan Cliff 5 दिवसांपूर्वी
My brain is now frazzled
naruto 577
naruto 577 5 दिवसांपूर्वी
João Matheus
João Matheus 5 दिवसांपूर्वी
SPIRAL OUT KEEP GOING oops, wrong spiral
Kurt Reber
Kurt Reber 5 दिवसांपूर्वी
The Primes were the original leaders of the autobots, who were benevolent, self-configuring robotic lifeforms. Whats all this about numbers?
Nida Zuberi
Nida Zuberi 5 दिवसांपूर्वी
The null risk reciprocally sin because romania intrestingly tempt to a milky call. juvenile, tested wheel
Shiraz Esat
Shiraz Esat 6 दिवसांपूर्वी
The summary at the end is so "spot on". It's humbling when you "discover" something that someone 200 years ago already figured, and also turned out to be half your age! But, it shows you still find numbers fun :)
Luke Cleland
Luke Cleland 6 दिवसांपूर्वी
because you're twirling them?
Dhruv Tyagi
Dhruv Tyagi 6 दिवसांपूर्वी
I lost you at 14:30
wow2fast4me 6 दिवसांपूर्वी
The null sponge jointly kill because wound lally cure with a wholesale ghana. terrible, hideous high bottom
Shawn Elliott
Shawn Elliott 6 दिवसांपूर्वी
7:10 - "And the reason we see only two of them when filtering for primes is that all prime numbers are either 1 or 5 above a multiple of 6, with the exceptions of 2 and 3." WHOA WHOA WHOA. You can't drop a fundamental fact like that without any explanation at all. _WHY_ are all primes equal to 6n + 1 or 6n + 5?
Релёкс84 6 दिवसांपूर्वी
It was explained just prior to that, starting at 5:45
Dave B
Dave B 6 दिवसांपूर्वी
*how do you make the computer graphics for thises videos? They are not just cgi but rather real plots and real numbers not like theres a function in photoshop to do this especially with thousands of numbers all at the correct location* ?
yves laflute
yves laflute 6 दिवसांपूर्वी
Totally irrational, the world is rational, no irrationals involved in reality. Have fun in the imaginary world. Volume of a sphere = exactly 5 times the base.
Релёкс84 6 दिवसांपूर्वी
That would be neat, but unfortunately it's just not correct.
Tom Collier
Tom Collier 6 दिवसांपूर्वी
Mother Nature is simple...she draws no straight lines. everything is expressed in a curved linear reciprocating processional hyperboloid. look into fibonacci...1235...leaves out 4, which is time and does not exist as a thing--theoria aphophosis-look him up.
Cube Fan
Cube Fan 6 दिवसांपूर्वी
Could we move on to third grade advanced calculus for the next video? Now where's the Tylenol???
Tom Bowen
Tom Bowen 6 दिवसांपूर्वी
awesome ......... but , 6 mins in and my ears are hurting with that "stupid spooky music" please let me know when there is a music free version as i would love to watch the rest , it's right up my lane , very sorry , pleaseeee and thanks , tom.
Aaron B.
Aaron B. 7 दिवसांपूर्वी
If you took the decimal away from pi (ie 314159...), would the infinite number be prime?
Релёкс84 6 दिवसांपूर्वी
It's wouldn't be a valid number, because integers can only be written with finite digits.
Schadrach42 7 दिवसांपूर्वी
Shouldn't there also be a point like that with 16 noticeable spirals? 16/pi is about as far from an even number of rotations as 6/pi is. Why doesn't that also seem to be the case?
Aristotle Stagirus
Aristotle Stagirus 7 दिवसांपूर्वी
Cool, but they can be arranged differently. Arranging things in different orders can give rise to interesting patterns which sometimes we can take advantage of. But, it should be taken into account that numbers can be arranged in very different ways with very different results. For example, why use base 10? Honestly the base 10 is not the most logical best base to use. We use it because of an arbitrary decision which is strongly related to the fact we have 8 fingers and 2 thumbs which means we can easily count to 10 with our two hands. But, historically humanity has used many other bases. A base which is 2^n where n is a natural number is better, i.e. 2, 4, 8, 16, 32, 64, etc. Of these, for our current level of intelligence, 8 is too simple and 32 is too complex, thus the base of 16 would likely be the best base. Another is, why plot these values in polar coordinates for the Cartesian coordinates (prime, prime). What about (prime, 1)? What about making patterns using Cartesian coordinates or spherical coordinates? Mind you, I am in no way saying there are not interesting things we can learn by forming numbers in a pattern like the one in this video, because clearly we can and that is a good thing. But, we should not forget to make other patterns and look for other interesting results from doing so.
Godspeedhero 7 दिवसांपूर्वी
Are you robot?
Godspeedhero 7 दिवसांपूर्वी
Mathematicians: "Mysteries of the universe? Hmph, there are no such things. Mathematics has all of the answers.... What was that? "Miillenium" what? Nonsense! I've never heard of such a thing!"
Godspeedhero 7 दिवसांपूर्वी
How did you make this video without mentioning Euler even one time? I'm amazed. I spoke 1min too soon. lmao Sorry for doubting you!
Dhruv Tyagi
Dhruv Tyagi 6 दिवसांपूर्वी
bravelittleroomba 7 दिवसांपूर्वी
"together with the angle that that radial line makes with the horizontal"
sam smith
sam smith 7 दिवसांपूर्वी
The boiling laundry supposedly file because close arespectively flow on a incompetent slime. eatable, stiff flag
robiandolo 7 दिवसांपूर्वी
Mickey Bitsko
Mickey Bitsko 7 दिवसांपूर्वी
Cool video. At first I thought, OMG! Symmetry among primes! Then you explained, and I quickly saw, that it was an illusion. Nice!
John Papish
John Papish 8 दिवसांपूर्वी
these spirals are really cool. wish i understood it. does this suggest that by extending these spirals way, way out we can identify really, really REALLY large prime numbers in a simple way ?
Релёкс84 8 दिवसांपूर्वी
This video mentions Dirichlet's theorem, which basically says that no, you cannot find primes more easily through this.
Elie Makhlouta
Elie Makhlouta 8 दिवसांपूर्वी
what if we used a number system based on pi rather than the decimal system?
kurt Mora
kurt Mora 8 दिवसांपूर्वी
The disgusted wallet biophysically choke because blowgun endoscopically impress below a oafish sled. eight, scarce share
Nisrasha 8 दिवसांपूर्वी
Will the prime ever be exactly 2pi
Nisrasha 8 दिवसांपूर्वी
@Релёкс84 Thank you
Релёкс84 8 दिवसांपूर्वी
No. Because if it ever did, the you would have found a way to write 2pi as a ratio of to integers. But that is impossible as 2pi is known to be irrational, so it will never happen.
lucabaar1 8 दिवसांपूर्वी
Doesn't it all have to do with our number system being 10-digit based? What would other number systems, with different iteration intervals, do to things like the mandelbrot set? Like counting in groups of 9 or 4, that would affect prime number sequence
Релёкс84 8 दिवसांपूर्वी
it has nothing to do with the way you write numbers. these are properties of the numbers themselves, independently of how you write them.
Mathew Smith
Mathew Smith 8 दिवसांपूर्वी
vortex maths. we learn this in junior school back on vulcan
R. K.
R. K. 9 दिवसांपूर्वी
22 Minutes of useless information, and yet, I am not mad having invested them.
Michals_ 9 दिवसांपूर्वी
I did not understand half of the math here but still find this incredibly interesting and fascinating
Rep0007 9 दिवसांपूर्वी
If you zoom the pattern out far enough, it forms the trademark stamp of UniCosmos Anachtakarg, the company that designed and manufactured our Universe in the first place (and many other Universes).
Sam Wang
Sam Wang 9 दिवसांपूर्वी
The chief sociology jointly transport because magician worrisomely agree underneath a lyrical branch. diligent, garrulous paul
Bernard Abrial
Bernard Abrial 9 दिवसांपूर्वी
Great video, thank you so much !
Immortal ZoDD
Immortal ZoDD 9 दिवसांपूर्वी
There are not an infinite amount of primes, and I think I can prove it
Релёкс84 9 दिवसांपूर्वी
@Immortal ZoDD While it's true that the "density" of prime numbers decreases the further you go (which is provable but harder to prove) that does not mean at all there is a finite number of them. Also, the sum of 1/n over all integers is not a finite number: it diverges to infinity. Just the fact that a sequence of numbers approaches zero *does not* mean its sum is finite, and this is a very important result. I'm not going to tell you the proof prime numbers are infinite just yet because it's much better for you if you work it out yourself, but I'll give you a clue on where to start: imagine you have a set which contains a finite number of prime numbers - it doesn't matter how many there are or which one they are, just asl ong as it's a finite group of primes. Now consider the number which is defined as the product of all those primes + 1. For instance, if your set contains 2, 3 and 5, this number will be 2x3x5 + 1 = 31. Now the question is: is this number prime? How can you tell? The answer to this question is not trivial, but finding it will definitely put you on the right path to understand why there are infinitely many primes. If you don't understand how to proceed I'll gladly give you the full reasoning but it's much better if at least you give it a try.
Immortal ZoDD
Immortal ZoDD 9 दिवसांपूर्वी
@Релёкс84 yeah, I meant a video explaining the proof, much like this one. I'm thinking of plotting prime number frequency, and I imagine at one point it will reach zero. I'm thinking it could be similar to how the sum of 1/n for n = 1 to infinity is a finite number
Релёкс84 9 दिवसांपूर्वी
@Immortal ZoDD Mathematicians don't prove things with videos. I don't know of a video about this, i just know the argument. But if you look up "infinite primes proof" on youtube you'll surely find a lot of results. I'd be glad to hear your reasoning though.
Immortal ZoDD
Immortal ZoDD 9 दिवसांपूर्वी
@Релёкс84 I bet there's video proof. Can you point me in the right direction? Want to see if my guess can hold any water
Релёкс84 9 दिवसांपूर्वी
it's easy to prove there is an infinite number of primes.
Axolition 10 दिवसांपूर्वी
**Looks at thumbnail** **Looks at title** Me: I don't know man, kinda looks like spiral energy from Gurren Lagan.
Caracho 10 दिवसांपूर्वी
12:40 this reminded me of a fingerprint. They also do have spirals. It´s always fascinating to discover mathematical structures and phenomena in nature
Gary Gallagher
Gary Gallagher 10 दिवसांपूर्वी
Man, your work is beautiful.
Edgar Jesus Moreno Castañeda
Edgar Jesus Moreno Castañeda 10 दिवसांपूर्वी
Apenas a los 6 minutos de vídeo me empeze a dar cuenta que no entiendo ni madres porque está en inglés
Alexandra Wimmer
Alexandra Wimmer 10 दिवसांपूर्वी
(I haven't watched the video to the end yet, and I should add that english isn't my language and that I have dyscalculia; Still I'm enormously fascinated by patterns! And I heard of Fibonacchi numbers and golden cut and golden angles and the nature artworkpatterns of Daysies, Sunflowers, Pinecones and at 4:03 your pattern beginns to look exactly like those! I was soo excitet ! I have tried to recreate such a spiral with a similar drawing like your polar coordinates radius circles and lines grid thingy and wondered why this pattern had anything to doo with numers ?? (1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13) BUT AT LAST I get to see the reason. Pictures say so much more than words (or numbers). MERCI BEAUCOUP to you !
PsyberSourcerer 10 दिवसांपूर्वी
Because primes exist at the gaps between multiples.
MLC Fonon
MLC Fonon 10 दिवसांपूर्वी
Thats why I have observed primes is a good tool for creating (musical) rythms ;)
Ritvars Rolis
Ritvars Rolis 10 दिवसांपूर्वी
Learning every day.Thank's!
Wanju Chien
Wanju Chien 10 दिवसांपूर्वी
PRIME number is the foundation of CREATION. The human MIND is constructed by Prime Number Matrix Grid. It is an unconscious KNOWING that NEW Ager refer ALL that is the PRIME CREATOR - INDEED. Currently we are at the 6th Prime, 13D and heading to the 7th Prime, 17D. Or it could be we already pass the 8th Prime 19D and heading to the 23D, 9th Prime in MIND. After that the path go backwards to the 7th PRIME and a heaven on earth will reflect on the OUTSIDE. IT is how TIME ( consciousness) moves.
Sujatha Ratnala
Sujatha Ratnala 11 दिवसांपूर्वी
Beautiful. It is interesting that number 6, 44 etc play a pivotal role. In polar plot of Natural numbers, 6 is close to 2pi with a bit of residue and hence the 6 streaks. And 6x + 1, 6x - 1 series are prime numbers and hence only 2 streaks of the 6 are prime.
Jake Keene
Jake Keene 11 दिवसांपूर्वी
@ 0:04 I immediately clicked out of the video.
Hugin Phosphor
Hugin Phosphor 11 दिवसांपूर्वी
1:03 why does the radius expand even tho it should stay at 3? puts the whole vid ad absurdum.
Kyeong woo Jeong
Kyeong woo Jeong 11 दिवसांपूर्वी
If it look like spiral, it is available transfer to sine wave ?
Mateo Aravena
Mateo Aravena 12 दिवसांपूर्वी
2:14 Jojo fans: hey, I know that guy!
Railgap Esoterica
Railgap Esoterica 12 दिवसांपूर्वी
One thing I learned from having a super-smart boss try to teach me algebraic topology is that these patterns are failry arbitrary and largely a product of how we choose to map / graph the data.
Michael Brown
Michael Brown 12 दिवसांपूर्वी
You inspire me sir!
Ignacio Roca
Ignacio Roca 12 दिवसांपूर्वी
4:08 my disappointment is immeasurable and my day is ruined. Why would you start it like that?
Timothy Agopian
Timothy Agopian 12 दिवसांपूर्वी
What if pi was a different value? Would this change anything? I have a way to calculate Pi with accuracy from the left to the right. I haven't gone more that 9 decimals deep but my understanding of what the value is, is: 3.141578444 Phi is: 1.617836463 I can go further but it takes a lot of space to write it out, if anyone could help me with writing a spreadsheet it could be done with a computer rather quickly...
G Kess
G Kess 13 दिवसांपूर्वी
Is there a way NOT to get spirals?
Mike Clarke
Mike Clarke 13 दिवसांपूर्वी
Why is 25 on the chart of primes?
Blanca Conejo
Blanca Conejo 13 दिवसांपूर्वी
Encause we are magnetically drawn to the core of Gods heart, three kingdoms pi and golden ratio are the order we left the order we go back joined is spiral infinity
Sophia Starks
Sophia Starks 13 दिवसांपूर्वी
This makes me like math for once..
R R 13 दिवसांपूर्वी
rocky or bullwinkle?
Daniel Marvin
Daniel Marvin 13 दिवसांपूर्वी
This man is jesus
Pumk!n 13 दिवसांपूर्वी
I dont have any interest in Math and I dont fully understand this, but its so Entertaining to watch, Thank you!
fglebre 14 दिवसांपूर्वी
A "totient" function, here written between totation marks, is not a totidian problem.
Valentin Berg
Valentin Berg 14 दिवसांपूर्वी
The word beautiful cant even describe how beautiful that shit is. This video made me feel like im high.
Great Bullet
Great Bullet 14 दिवसांपूर्वी
The zero is actually creating primes ! Diabolic decimal system :)
Релёкс84 8 दिवसांपूर्वी
The existence and value of primes is competely independent from the way you write numbers.
Syracax 14 दिवसांपूर्वी
they dont
Deepa Rao
Deepa Rao 14 दिवसांपूर्वी
Awesome explanation 👍🏼❤️
Rumple Stilskin
Rumple Stilskin 14 दिवसांपूर्वी
Any chance you can do me a huge favor? I need a visualization of a base 3 number system prime number spiral. Show me prime numbers in a projection, using a base 3 system. =)
Релёкс84 13 दिवसांपूर्वी
The base you write them in doesn't change how it looks. That's because you draw them, not write them.
Дмитрий Ляпин
Дмитрий Ляпин 15 दिवसांपूर्वी
I have two questions... I wrote small program to find cases when the result of the next step is closer to a integer number of full turns than ever before.That's what I've got. 1 radian is about 0 full turns, the difference is 1.0 6 radians is about 1 full turn, the difference is -0.28318530717958623 19 radians is about 3 full turns, the difference is 0.1504440784612413 25 radians is about 4 full turns, the difference is -0.13274122871834493 44 radians is about 7 full turns, the difference is 0.017702849742896376 333 radians is about 53 full turns, the difference is -0.008821280518098717 710 radians is about 113 full turns, the difference is 6.028870677710074E-5 103993 radians is about 16551 full turns, the difference is -1.912933227998792E-5 312689 radians is about 49766 full turns, the difference is 2.9007205739617348E-6 1980127 radians is about 315147 full turns, the difference is -1.725052491963197E-6 2292816 radians is about 364913 full turns, the difference is 1.1757947504520416E-6 4272943 radians is about 680060 full turns, the difference is -5.497234027984632E-7 Well, I got 1, 6, 44 and even 710, the numbers of steps which about you spoke But 19-, 25- and 333-groups of spirals just become lost among the others. T hat's why I have those two questions. 1) Which spirals would we see? 2) Is set of these numbers endless? P.S. I'm sorry for my English, I am Russian :) I would be grateful if you could check me :)))
Релёкс84 13 दिवसांपूर्वी
Yes this sequence of numbers is endless, because you can get as close as you want to pi using rational numbers, but never reach it since it's irrational.
Thanakrit Gu
Thanakrit Gu 15 दिवसांपूर्वी
kudos to the animator of this video
EdiKurn_ 15 दिवसांपूर्वी
We can draw use prime number, see my video
Marq 15 दिवसांपूर्वी
Phenomenal. Perfect. Pleasantly presented.
Connie Prude
Connie Prude 15 दिवसांपूर्वी
The frequent grill rhetorically use because front strangely calculate above a abusive example. gigantic, grubby gruesome bridge
Релёкс84 13 दिवसांपूर्वी
Man, they are here too?
Steve Kirkby
Steve Kirkby 15 दिवसांपूर्वी
Ah, damn, just had it thrown down my throat that I am thick as pig poo. I thought I was, sort of, moderately bright.... then got 5 minutes into this and just zoned. Jeez.
Neth Winkles
Neth Winkles 16 दिवसांपूर्वी
The calm adapter electrophysiologically support because season preclinically peep behind a utter cupcake. crowded, voracious drill
Nicol Bolas
Nicol Bolas 16 दिवसांपूर्वी
Hmm doesn't this mean you solved fermats last theorem?
Martín Elosú Dietz
Martín Elosú Dietz 17 दिवसांपूर्वी
So... If you zoom out even more, you will get new spirals everytime, and each time they will be less "spiraled", meaning you got a new rational approximation for π that is more accurate than the one before..? Right? *mindblow* EDIT: When zooming tends to infinity, you would spect to visualize perfectly straight lines that arise from the origin, as you got an infinite rational approximation to π
But how does bitcoin actually work?
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