# Why do prime numbers make these spirals?

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**Published: 08 October 2019**- A story of mathematical play.

Home page: 3blue1brown.com

Brought to you by you: 3b1b.co/spiral-thanks

Based on this Math Stack Exchange post:

math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers/885894

Want to learn more about rational approximations? See this Mathologer video.

cepnet.mobi/video/CaasbfdJdJg/video.html

Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video:

cepnet.mobi/video/iFuR97YcSLM/video.html

Dirichlet's paper:

arxiv.org/pdf/0808.1408.pdf

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.

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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/album/the-music-of-3blue1brown

Stream the music on Spotify:

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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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3Blue1Brown6 days backImportant 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.

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

101 x 20 = 2020. 101 is a very interesting prime, I think. At least it is palindrome

@Joe Banks 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.)

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!

my question is this, this points are on one plane of a circle, or i guess saying it's represented in 2D, what would this look like represented as a 3D sphere?

"He who holds the secret of the seven stars in his right hand, and draweth the double sword from his mouth, and his countance was as the sun shineth in his strength"

Revelations

seriously, at least for a moment, you brought me tears in the end.

and each time I see a new video from 3b1b it just keep getting better interms of animation, content and explanation. If GRANT you are reading this, remeber that you have changed the way people look at math...

always greatful for that

"SWING ON THE SPIRAL.....", "SPIRAL OUT KEEP GOING......"

mjk......

Biostatistics, so mix it! The Bible says that the first team in the end. But two bitcoinb are dead!

It IS beautiful!! But..

eventually that's no business of prime numbers!

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

I know this is supposed to be a close to layman video, but as someone who did math olympiads it hurts to hear someone refer to residue classes, mod m, Euler's totient function etc as overly fancy, unnecessary jargon D:

Why don't most of you look up Plasma/electric Universe. Explains everything

I love the sentiment at the end, about stumbling upon something on your own before encountering it academically. The first time I did this, I felt like I needed to tell everyone - not as if the idea belonged to me, but just because it felt so exciting, almost like a validation of the pursuit of learning. I'm fairly sure that was a huge part of what "flipped the switch" for me and made me interested in learning in my last year of high school. Probably never would've went to college if not for that moment.

I did not deserve to watch this for free.

so you gonna make virtual spiral printed images that are circle shapped? thats nice one bro

I absolutely LOVE how popular this video is. It gives me hope for humanity, after all.

could you do a video explaining the Foucault pendulum and why the rate of rotation of the Earth times the sine of the number of degrees of latitude.

旧2ch(5ch)数学板フェルマー最終定理についてのスレ主です。掲示板の内容話題にするの許しました。

しかし、これ凄いですね。

しかし、パソコン使うのは宇宙の法律違反なの知ってましたか!！?？

あなたこのままいくと全て見付けてしまいますよ。何の思い出もなく。

将棋の動画載せて下さいよ。

本当に強いのかハチワンダイバーと言う漫画のザンガヘッドで調べます。

I asked my teacher: "when Am I going to use maths?"

And my teacher responded: "Anytime you find it difficult sleeping just try using it" 😂😂😂

I was finding it difficult sleeping until I came across the video. 😅😅😅

Pi is exactly 3...

I have no idea how you made this animation, but it is so smooth it looks incredible

That moment at 2.26, when the music changed, and when the pattern appeared, almost brought tears to my eyes. Such a beautiful layout of this extraordinary symmetry. I propose 3blue1brown for the academy award, the best director of the year.

To add on this, the moment from 4:00 to around 4:02, I'm okay with the spirals and the rays, but did you guys see that crazy flower-like formation emerging from the zooming out?

Can you prove that the smallest possible value of n m | PI - n/m | = ~0.01 for any two natural numbers n and m? Namely for n=355 and m=113. No other positive integers, no matter how large, gives something smaller. Or?

Time to see a psychologist... wooo!

Очень интересно, жаль на русском таких видео почти не найти

Make a video for logarithm

Its real fun and enjoy to watch your video's

Can you please plot the same graph but instead of 1, 2, 3,4, 5,6 use "1/3π, 2/3π, π, 4/3π, 5/3π, 2π" ? My guess is spirals all go bye-bye? But at least then the prime numbers may reveal something interesting.

answer: they don't... well, they do if you chart growing numbers around a radial grid in a specific way. But if you pick a different arbitrary spacing, then you will find they don't.

7:20 so it's mod6, with just 1 and 5 as "generators"?

that reminds me about Bach-Leibniz music system,

which is mod12, with 1 and 11, plus 5 and 7, as generators

there is a whole playlist about that, hope someone could link it back here

9:09 which are the "generators" here?

https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n

There is prime generator, also a function which for sure get another prime... It's not hard... It's golden ratio with multiplication, than if you cut it by root or something it looks like good ratios too maybe....

And there is a generator which yields every prime. And also there is relation between those two sets of numbers. It's really simple.

Everything in this video is amazing, I've just got hypnotized by watching it. Tyvm for making it.

Wow, what you said at the end of the video made me realize why I'm so good at learning new things in math. I'm always playing with math, experimenting with the "fun" parts of topics. Prime numbers, sequences, modular arithmetic, rational approximation, number theory, combinatorics, digits of irrational numbers, numeral systems, differential systems, and fractals/chaos are generally what intrigue me, and they seem to show up everywhere, lol!

And I must say that statistics is the worst kind of "math" in existence. Let it be a science, not a subject of math.

Really good and illustrative video!

At the beginning I thought it's some kind of illuminati conspiracy, but you explained the reasons for this plotting very well!

pi is 3

e is 3

3 is 3

life is simple.

It could be universal code? I notice star trails have a very similar pattern and it’s all over the universe.

This video reminds me of the time when I read that 196,883 + 1

= 196,884.

My friend, your videos are so filled with wonder and fun, that I wish I'd kept with my math PhD rather than my current pursuit. Ah, well!

It'd be nice to have a video about "lay research" mathematics, and how one could "contribute".

I don't even like math

Hello im enjoy

Those patterns don't look very pointless to me. More like pointyful...

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

When i was in 11th grade i figured out the primes followed 5k+6, 7k+6 kind of sequence (i had to think about this bcs of my exam failure) and noticed there was some weird pattern. Appently i was right i guess. But still that damn exam

Man, if my math class played a video related to the topics we were studying in a similar fashion before giving the lecture, I would've been so much hyped for it.

The practical and professional utility of understanding maths aside, videos like this highlight just how visceral the subject can be, and worth studying simply for curiosities sake. As someone who has long lamented my ignorance of the subject, thank you for increasing my motivation to do something about it!

Now play the pattern through morse code, the Illuminati wishes to speak with you.

Someone chose to plot numbers in a way that made a pretty pattern and was then surprised that a subset of these numbers made a modified version of this pattern... Hooplidooda!

Damn!!!! I want him as my tutor.

I dont understand a thing but im thoroughly enjoying this video

What makes those histograms and not bar charts? I thought histograms varied in bar width and used area to represent data?