Why do prime numbers make these spirals?

8 okt. 2019
2 781 030 Weergaven

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:
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.
nlworld.info/key/video/pJmSpcfJm4GVgJ4
Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video:
nlworld.info/key/video/yn6mhJ6akJqEgoQ
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:
open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
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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Reacties
  • But Pi is an inexact number. So much for the mathematical perfection of the universe...

    Hogvlog Films: SailingHogvlog Films: Sailing7 uur geleden
    • What do you even mean by "inexact number" ?

      Релёкс84Релёкс843 uur geleden
  • Would also like to see the points plotted in cylindrical coords like (2,2,2), (3,3,3),(5,5,5) etc. Those spirals would curl up like smoke from a fire.

    planet42planet4220 uur geleden
  • I was just learning about LRAM in my calculus class, and it came up when he mentioned using calculus with functions using complex numbers.

    Jordan WibleJordan WibleDag geleden
  • This is the best way I've heard this described

    Julius CaesarJulius Caesar2 dagen geleden
  • I do not know where to ask this question. I noticed a peculiar mathematical phenomenon, while looking at the calculation of 662314 days with different length years (don't ask). The math is 662314/359 = 1,844.8857938718662952646239554318, which is 1844 and 318 days. That is, 0.8857938718662952646239554318 x 359 = 318 days. Further, 0.8857938718662952646239554318 x 360, instead of 359, is 318.88579387186629526462395543175. The exact same decimal. The difference at the end is simply due to the places my calculator can produce. That is, we can do this into infinity. I noticed that the result is due to the relationship between 359 and 360, or any odd number and that number plus 1. For instance, if we take 663840 days, which is 1844 x 360, and divide it by 359, we get 1,849.1364902506963788300835654596, which is 1849 years and 49 days. The remainder of 0.1364902506963788300835654596 x 360 is 49.136490250696378830083565456.This means it is not the large number we start with that creates this phenomenon. If we take 662314 and divide it by three, we get 220,771.33333333333333333333333333. The remainder multiplied by 3 results in 1. If it is multiplied by 4, it results in 1.3333333333333333333333 I do not know what this rule is, since without a name for it, it is very difficult to look up on Google.

    Theodore TurnerTheodore Turner2 dagen geleden
  • I'd like to see the pattern in primes if there were any. But there are none except all prime waves cross at zero. Light waves of different lengths, all starting at zero and going off into infinity is the way to think of prime numbers. I call them prime waves. They go both directions from zero, positive and negative. So for example the number 3. It would generate a prime wave that crosses the number line at zero, then 3, then 6, then 9 and so forth out into infinity. The thing about primes is you have to travel the wave until it crosses off all of the numbers in it's path as it crosses the number line. Each new prime number starting a new prime wave and all of the prime waves combined traveling out from zero until they don't cross out a number on the number line. And this new prime becoming a new prime wave traveling out with the other prime waves at the speed of light crossing out all of the multiples of the existing prime numbers until the next prime number is found. Our speed is the speed of light. Thus you could calculate how far the prime waves traveling at the speed of light would travel in a year. And thus how many new primes can theoretically be found per year using the speed of light as the basis for calculation.

    James ColeJames Cole3 dagen geleden
    • I might add that the distance between numbers can be any size. So it would be the speed of our fastest computer that determines the speed of our lightwaves.

      James ColeJames Cole3 dagen geleden
  • The Eratostene's filter leaving empty field like the galaxies arms

    hgjf3hgjf33 dagen geleden
  • Yeah I don't see any new revelations here about pi or primes. When I was a kid in school pi was this mysterious number that was used to calculate circles. Yet it is so easy to make pi something that even a small child can grasp. If you calculate the circumference of a square, lets say each side is 1, then to go around that square is 4. Now set a circle in that square and the circumference is 3.14 or pi. So the ratio is 4 to 3.14 The ratio of the circumference of a square to a circle's circumference that fits inside that square. And really that ratio is not really just 4 to 3.14. It is also the ratio of one side of the square or 1 to one side of the circle, or 1/4th of the circle or .785. So 1 to .785, that is the exact same ratio as pi.

    James ColeJames Cole3 dagen geleden
  • You were doing an excelled job of explaining until you got the the primes but I see it now. The primes are doubled also. So the prime number 3 would be (3, 3). 5 would be (5, 5) and so on.

    James ColeJames Cole4 dagen geleden
  • 🥱😴

    W0LFW0LF4 dagen geleden
  • I usually hear this called aliasing.

    Christopher CrawfordChristopher Crawford4 dagen geleden
  • Am I the only one who saw Veritasium in the Patreon list?

    Nisheet VermaNisheet Verma4 dagen geleden
  • like your videos, thk!

    Rex WangRex Wang5 dagen geleden
  • Well no shit!

    Stewart PerthouStewart Perthou5 dagen geleden
  • The panoramic low indirectly rub because psychiatrist microbiologically identify for a acceptable side. free, flashy slime

    sam smithsam smith5 dagen geleden
  • This is the dumbest thing i've ever seen.... Seriously ppl..... The first thing they do is write the numbers in a spiral pattern then afterwards marvel at the fact that if you remove some numbers it still looks like a spiral.

    Peter Gram MortensenPeter Gram Mortensen7 dagen geleden
    • I like your simplicity of a solution.

      James ColeJames Cole3 dagen geleden
  • Are there primes in all different number bases?

    John AldenJohn Alden7 dagen geleden
    • Primes are independent of the base or whatever system you use to write them with.

      Релёкс84Релёкс847 dagen geleden
  • So, if someone plans to animate hyperspace, they can just take prime numbers!

    Marah WolsfeldMarah Wolsfeld7 dagen geleden
  • Beautiful 👍

    ayush anandayush anand8 dagen geleden
  • Beautifully visualized, clearly explained. Bonus: the plug for advanced education toward the end.

    David SyskoDavid Sysko9 dagen geleden
  • Wormhole.

    LetysLetys9 dagen geleden
  • The fact that 2.7 million people watched this gives me some hope for mankind.

    Christopher MaupinChristopher Maupin9 dagen geleden
    • Take a cold shower

      W0LFW0LF4 dagen geleden
  • Easy Fix: Write the numbers in a line!

    DuesternisDuesternis9 dagen geleden
  • Those spirals are fascinating and the explanations are amazing.

    Sun DaojunSun Daojun9 dagen geleden
  • 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.

    James the BearJames the Bear10 dagen geleden
  • Dave Rubin. You sound like Dave Rubin. I've been trying to place it for weeks lol

    Michael KilgoreMichael Kilgore10 dagen geleden
  • The Creator(of the universe) is an organised artist

    Julie JohnsonJulie Johnson10 dagen geleden
  • 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 SipilainenJari Sipilainen10 dagen geleden
  • 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!!!

    Jari SipilainenJari Sipilainen10 dagen geleden
  • Dude, you make me love Maths more and more every day

    Shreya GokhaleShreya Gokhale10 dagen geleden
  • 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.

    joel ivan cittarjoel ivan cittar11 dagen geleden
  • 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.

    Jeff MurphyJeff Murphy11 dagen geleden
  • 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$

    BrunoBruno11 dagen geleden
  • You can make a religion out of this.

    NubyNuby11 dagen geleden
  • The unusual kendo ordinarily bang because ray bailly whisper during a irate baboon. terrific, bloody parallelogram

    Hunter WatkinsHunter Watkins11 dagen geleden
  • math hasn't made me this happy since 5th grade

    Ellie MäkeläEllie Mäkelä12 dagen geleden
  • Everyone who came here in this video are knowledge seekers.

    Dip BhowmikDip Bhowmik12 dagen geleden
  • Wow

    CANNAHEALTHCANNAHEALTH12 dagen geleden
  • i,m 844 mos. old ,4 goes into ,=211 see how it all works out.....hmmmm

    thomas dunhamthomas dunham12 dagen geleden
  • 13:52 Oh gosh I think I just got hallucinations from that

    Alex HandoAlex Hando12 dagen geleden
  • My brain is now frazzled

    Alan CliffAlan Cliff13 dagen geleden
  • Amazing...

    naruto 577naruto 57713 dagen geleden
  • SPIRAL OUT KEEP GOING oops, wrong spiral

    João MatheusJoão Matheus13 dagen geleden
  • The Primes were the original leaders of the autobots, who were benevolent, self-configuring robotic lifeforms. Whats all this about numbers?

    Kurt ReberKurt Reber13 dagen geleden
  • The null risk reciprocally sin because romania intrestingly tempt to a milky call. juvenile, tested wheel

    Nida ZuberiNida Zuberi13 dagen geleden
  • 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 :)

    Shiraz EsatShiraz Esat13 dagen geleden
  • because you're twirling them?

    Luke ClelandLuke Cleland13 dagen geleden
  • I lost you at 14:30

    Dhruv TyagiDhruv Tyagi13 dagen geleden
  • The null sponge jointly kill because wound lally cure with a wholesale ghana. terrible, hideous high bottom

    wow2fast4mewow2fast4me13 dagen geleden
  • 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?

    Shawn ElliottShawn Elliott14 dagen geleden
    • It was explained just prior to that, starting at 5:45

      Релёкс84Релёкс8413 dagen geleden
  • *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* ?

    Dave BDave B14 dagen geleden
  • 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.

    yves lafluteyves laflute14 dagen geleden
    • That would be neat, but unfortunately it's just not correct.

      Релёкс84Релёкс8414 dagen geleden
  • 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.

    Tom CollierTom Collier14 dagen geleden
  • Could we move on to third grade advanced calculus for the next video? Now where's the Tylenol???

    Cube FanCube Fan14 dagen geleden
  • 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.

    Tom BowenTom Bowen14 dagen geleden
  • If you took the decimal away from pi (ie 314159...), would the infinite number be prime?

    Aaron B.Aaron B.14 dagen geleden
    • It's wouldn't be a valid number, because integers can only be written with finite digits.

      Релёкс84Релёкс8413 dagen geleden
  • 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?

    Schadrach42Schadrach4214 dagen geleden
  • 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.

    Aristotle StagirusAristotle Stagirus14 dagen geleden
  • Are you robot?

    GodspeedheroGodspeedhero15 dagen geleden
  • 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!"

    GodspeedheroGodspeedhero15 dagen geleden
  • 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!

    GodspeedheroGodspeedhero15 dagen geleden
    • Haha!!

      Dhruv TyagiDhruv Tyagi13 dagen geleden
  • "together with the angle that that radial line makes with the horizontal"

    bravelittleroombabravelittleroomba15 dagen geleden
  • The boiling laundry supposedly file because close arespectively flow on a incompetent slime. eatable, stiff flag

    sam smithsam smith15 dagen geleden
  • wormhole?

    robiandolorobiandolo15 dagen geleden
  • Cool video. At first I thought, OMG! Symmetry among primes! Then you explained, and I quickly saw, that it was an illusion. Nice!

    Mickey BitskoMickey Bitsko15 dagen geleden
  • 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 ?

    John PapishJohn Papish15 dagen geleden
    • This video mentions Dirichlet's theorem, which basically says that no, you cannot find primes more easily through this.

      Релёкс84Релёкс8415 dagen geleden
  • what if we used a number system based on pi rather than the decimal system?

    Elie MakhloutaElie Makhlouta15 dagen geleden
  • The disgusted wallet biophysically choke because blowgun endoscopically impress below a oafish sled. eight, scarce share

    kurt Morakurt Mora15 dagen geleden
  • Will the prime ever be exactly 2pi

    NisrashaNisrasha16 dagen geleden
    • @Релёкс84 Thank you

      NisrashaNisrasha15 dagen geleden
    • 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.

      Релёкс84Релёкс8415 dagen geleden
  • 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

    lucabaar1lucabaar116 dagen geleden
    • it has nothing to do with the way you write numbers. these are properties of the numbers themselves, independently of how you write them.

      Релёкс84Релёкс8415 dagen geleden
  • vortex maths. we learn this in junior school back on vulcan

    Mathew SmithMathew Smith16 dagen geleden
  • 22 Minutes of useless information, and yet, I am not mad having invested them.

    R. K.R. K.16 dagen geleden
  • I did not understand half of the math here but still find this incredibly interesting and fascinating

    Michals_Michals_16 dagen geleden
  • 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).

    Rep0007Rep000716 dagen geleden
  • The chief sociology jointly transport because magician worrisomely agree underneath a lyrical branch. diligent, garrulous paul

    Sam WangSam Wang16 dagen geleden
  • Great video, thank you so much !

    Bernard AbrialBernard Abrial16 dagen geleden
  • There are not an infinite amount of primes, and I think I can prove it

    Immortal ZoDDImmortal ZoDD17 dagen geleden
    • @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.

      Релёкс84Релёкс8416 dagen geleden
    • @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

      Immortal ZoDDImmortal ZoDD16 dagen geleden
    • @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.

      Релёкс84Релёкс8416 dagen geleden
    • @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

      Immortal ZoDDImmortal ZoDD16 dagen geleden
    • it's easy to prove there is an infinite number of primes.

      Релёкс84Релёкс8416 dagen geleden
  • **Looks at thumbnail** **Looks at title** Me: I don't know man, kinda looks like spiral energy from Gurren Lagan.

    AxolitionAxolition17 dagen geleden
  • 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

    CarachoCaracho17 dagen geleden
  • Man, your work is beautiful.

    Gary GallagherGary Gallagher17 dagen geleden
  • Apenas a los 6 minutos de vídeo me empeze a dar cuenta que no entiendo ni madres porque está en inglés

    Edgar Jesus Moreno CastañedaEdgar Jesus Moreno Castañeda17 dagen geleden
  • (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 !

    Alexandra WimmerAlexandra Wimmer17 dagen geleden
  • Because primes exist at the gaps between multiples.

    PsyberSourcererPsyberSourcerer18 dagen geleden
  • Thats why I have observed primes is a good tool for creating (musical) rythms ;)

    MLC FononMLC Fonon18 dagen geleden
  • Learning every day.Thank's!

    Ritvars RolisRitvars Rolis18 dagen geleden
  • 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.

    Wanju ChienWanju Chien18 dagen geleden
  • 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.

    Sujatha RatnalaSujatha Ratnala18 dagen geleden
  • @ 0:04 I immediately clicked out of the video.

    Jake KeeneJake Keene18 dagen geleden
  • 1:03 why does the radius expand even tho it should stay at 3? puts the whole vid ad absurdum.

    Hugin PhosphorHugin Phosphor18 dagen geleden
  • If it look like spiral, it is available transfer to sine wave ?

    Kyeong woo JeongKyeong woo Jeong19 dagen geleden
  • 2:14 Jojo fans: hey, I know that guy!

    Mateo AravenaMateo Aravena19 dagen geleden
  • 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.

    Railgap EsotericaRailgap Esoterica19 dagen geleden
  • You inspire me sir!

    Michael BrownMichael Brown19 dagen geleden
  • 4:08 my disappointment is immeasurable and my day is ruined. Why would you start it like that?

    Ignacio RocaIgnacio Roca19 dagen geleden
  • 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...

    Timothy AgopianTimothy Agopian20 dagen geleden
  • Is there a way NOT to get spirals?

    G KessG Kess20 dagen geleden
  • Why is 25 on the chart of primes?

    Mike ClarkeMike Clarke20 dagen geleden
  • 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

    Blanca ConejoBlanca Conejo20 dagen geleden
  • This makes me like math for once..

    Sophia StarksSophia Starks20 dagen geleden
  • rocky or bullwinkle?

    R RR R21 dag geleden
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