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Why do prime numbers make these spirals? (2019)
- zmmmmm 3y agoBit of a let down when they reveal that plotting integers in general results in the same spiral pattern as primes. So naturally primes as a subset of integers also produce spirals. Seems the title is a bit misleading!
- desertraven 3y agoA bit misleading I agree. I suppose since prime numbers DO make these spirals, it's not inaccurate.
- smitty1e 3y agoWe expect patterns from quotidian integers. The seeming chaos of primes, in contrast, brings the intrigue.
- Bytewave81 3y agoIt's referencing the Math Stack Exchange question which asked the same question: https://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers/885894 https://math.stackexchange.com/questions/885879/meaning-of-r...
- crdrost 3y agoI don't know, for me the fact that this gives you a visualization of a totient is really interesting. Another really interesting idea is to deliberately change the thing which you are rationally approximating; you don't have to rationally approximate π if you don't want to, that's just if you make steps of 1 radian. Make steps of q radians and you get the denominators for rational approximations of q/π. This is used in the golden spiral algorithm[1] to evenly-ish distribute points on a sphere, we choose the most irrational number q/π = φ, the golden ratio. Since all of its rational approximations suck, the spirals are as inoffensive as they can be. 1. https://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere https://stackoverflow.com/questions/9600801/evenly-distribut...
- subroutine 3y agoSomething less interesting is testing what the polar plot looks like if you plot the angle in degrees instead of radians. Or, like in this plot, where I defined 10 degrees as exactly one complete turn around the circle: https://i.ibb.co/59k5dRT/polarplot.png https://i.ibb.co/59k5dRT/polarplot.png Slightly more interesting is what the primes look like when one complete turn is defined as 1, 2, 3... 30 units: https://i.ibb.co/F0GvrSY/polarplot2.png https://i.ibb.co/F0GvrSY/polarplot2.png
- onethought 3y agoThe whole point of the video is that asking these kinds of questions even if at first they seem kind of silly or arbitrary often lead to really interesting observations. In this case: Dirichlet's theorem on arithmetic progressions [0] [0] https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith...
- laserbeam 3y agoI found it insightful, not a let down. It says a lot about human nature and our biases towards spotting patterns that may be incomplete or inaccurate.
- euroderf 3y agoIn the absence of reliable telepathy, there is indeed survival value in spotting patterns in incomplete information.
- desertraven 3y agoWhat's the commentary on prime numbers currently? Is it expected that there is some unfound pattern to them? Or will they always remain elusive?
- timetraveller26 3y agoYes
- frogpelt 3y agoNo.
- Taywee 3y agoMaybe.
- C-x_C-f 3y agoElusiveness is subjective, but there's certainly plenty of patterns to be found (though, to be fair, what constitutes a pattern might also be subjective). You should check out Terence Tao's blog [0], he's not only one of the most prominent mathematicians in the field but he's also excellent at explaining things. These slides [1] for example don't require any background in math. [0] https://terrytao.wordpress.com https://terrytao.wordpress.com [1] https://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf https://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf
- smackeyacky 3y agoPrime numbers are important for cryptography - if there is a pattern to them it might have an effect on what we think is secure or not. https://www.abc.net.au/news/science/2018-01-20/how-prime-numbers-rsa-encryption-works/9338876 https://www.abc.net.au/news/science/2018-01-20/how-prime-num...
- mathgenius 3y agoThe answer to these questions is the Riemann hypothesis. John Baez just wrote a related paper about "motives" for beginners (undergrads?) [1]. It's worth a read even if you squint at the equations like i do, there's plenty of interesting commentary as well. [1] https://math.ucr.edu/home/baez/motives.pdf https://math.ucr.edu/home/baez/motives.pdf
- harles 3y agoIt could use a (2019) in the title. Interesting, but not recent.
- irrational 3y agoDoes it matter for math articles? It’s not like the math has changed in 4 years.
- dylan604 3y agoSomething has to define when new math came to be /s
- mypornaccount 3y agoI double checked, 1+1 is still 2 in ios 16.3.1
- interestica 3y agoSometimes it's hardware https://en.m.wikipedia.org/wiki/Pentium_FDIV_bug https://en.m.wikipedia.org/wiki/Pentium_FDIV_bug
- jagged-chisel 3y agoIn 16.4.1, you have the opportunity to make it 4.15% bigger.
- varjag 3y agoOn the other hand 1+2+3 is 24 in iOS 11.1
- edgineer 3y agoThere is an aspect of novelty when it comes to YouTube videos
- thrdbndndn 3y ago> Does it matter for math articles It matters for all the articles, so yes.
- deleted 3y ago[deleted]
- m3kw9 3y agoI can also make all integers look like sine waves sin(x)
- mlyle 3y agoBetter divide x by something or you get a pretty crummy sine. Or multiply by 19.
- quickthrower2 3y agoOr use the high-school sin. (Take's degrees as input)
- b33j0r 3y agoy = (-1)**x -1 is a circle, brah!
- MayeulC 3y agoTangentially, this is directly related to the article, as x*sin(x) is the x-component of plotting (x,x) in polar coordinates: with more precision (with x a real), you'd obtain a single spiral, but with integers where 2pi≈6, they appear as six. Plotting primes p like this is plotting p*e^(ip), but the spirals are an interesting observation. You can straighten them out by adding a pi factor to the polar coordinate (so iside the sin...) but that's less interesting.
- gnarbarian 3y agoI suspect most functions would make a spiral on a polar graph if you zoomed out far enough.
- onethought 3y agoWell, except as the video points out - primes don't make spiral when you zoom out far enough, they make a "rays" pattern... which then if you zoom out even further becomes a much gentler spiral.
- jameshart 3y agoAnd the fact that pi is irrational means you will continue zooming out getting ever closer to rays, but never actually find rays - you’ll always have a slight spiral.
- smitty1e 3y ago> These patterns are certainly beautiful, but they don't have a hidden, divine message about primes. As far as we know, for now.
- jacknews 3y agoYeah, it's clearly someone's fingerprint: https://i.imgur.com/yFMPmtW.png https://i.imgur.com/yFMPmtW.png
- rarx 3y agoa factor, plus one. maybe? so it’s the pair of primes in that short sequence
- Scubabear68 3y agoI’m hardly a math genius, but isn’t this basically because he’s using polar coordinates, where nearly any trend is going to look like curves and spirals?
- hn_throwaway_99 3y agoHence, the video...
- CyberDildonics 3y agoHence the video what? It's polar coordinates making the spiral, not prime numbers.
- westcort 3y agoThe primes >3 are all of the form 6k+1 or 6k-1, so they are close to 6k. As k increases, the resulting marks on the polar coordinate system form nearly complete turns because 6 is close to 2 pi. Hence, spirals. See my other comment for a clearer explanation of why primes >3 all must necessarily be one more or one less than a factor of 6k where k is an integer. That is the crucial piece that the author did not make totally clear.
- tomxor 3y agoThis is pointed out in the video
- hn_throwaway_99 3y ago
- hn_throwaway_99 3y agoI thought this video was fantastic. Just a great intro and example of how "playing" with numbers and visualizations can help you understand deeper, more profound concepts. For commenters saying "well, doesn't anything in polar coordinates end up like spirals?", do yourself a favor and watch the video. He says as much pretty early on that yes, any plotting of the integers like that will look like spirals, but goes into excellent detail in my opinion explaining why the specific patterns you see with prime numbers are as they are.
- westcort 3y agoOne of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is of the form 6k+1, 6k-1, 6k+2, 6k-2, 6k+3, or 6k-3. But you cannot make primes with the numbers of the form 6k+2, 6k-2 (they would always have to be divisible by 2), and you cannot makes primes with numbers of the form 6k+3, 6k-3 because they are always divisible by 3. So what are you left with? All primes >3 must of the form 6k+1 or 6k-1. And that factor 6 is just a bit less than 2 pi (a full turn in radians) so you get spirals from the offset. They are also a pixel or two off, but that is imperceptible. The same logic is a nice exercise to apply to the problem of why primes >2 can only be of the form 4k+1 or 4k-1. Apply the same logic as above.
- pcmaffey 3y agoYes, it was such a thrill to discover this in college (not a math major). I formulated it as all primes >2 are “factors” of 1 and 5 in a base 6 notation.
- MayeulC 3y agoWell, factor is clearly the wrong technical term here, you probably meant the last digit? Like the last digit (remainder of modulo 10 division: p%10 in C-notation) of a prime number (p>5) in base 10 can be {1,3,7,9}. And yes, in base 6 that becomes 1 and 5, as p%6 = 1 or 5, equivalent to p=6k±1. Interesting observation, thanks :)
- deleted 3y ago[deleted]
- CGamesPlay 3y agoThis is neat and I never noticed this. For 5, I guess the linear component would be (2*3*5)k, but it isn't as interesting or useful because the constant component would be +/- 1, 7, 11, or 13. This method feels like it's basically a "higher order prime sieve". [append] Oh, and because this pattern (of 1 always being one of the constants) carries out for arbitrarily large linear coefficients, that also explains the "twin prime" phenomenon: https://www.youtube.com/watch?v=QKHKD8bRAro https://www.youtube.com/watch?v=QKHKD8bRAro
- ShaurAsar 3y ago[dead]
- peter_retief 3y agoThis is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?
- mik1998 3y agoWe don't know whether it's true.
- qumpis 3y agoWell, then it's a great mystery of up to 10^18: "T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 10^18"
- peter_retief 3y agoIt is true but unproven.
- misja111 3y agoThe odd number sums follow from the fact about the even numbers.
- peter_retief 3y agoWhy though?
- misja111 3y agoIf you can make any even number with 2 primes, you can make an odd number by subtracting one odd prime from the odd number first (e.g. 3) and the resulting even number with 2 more primes.
- Poppys 3y agoMy browser did not enjoy that 40MB web page.
- aaron695 3y ago[dead]
- kfrzcode 3y agoIt's because we live inside a spiral galaxy that's transitioning into a new galaxy type so we see through a spiral
- tabtab 3y agoDoesn't work in Australia.
- anoncow 3y agoCould there be a way to derive prime numbers based on an extrapolated graph which would be less resource intensive as compared to current prime identifying methods?
- y42 3y agoShameless self-promotion, just because it seems to fit: I build an "animation engine" around primes and spirals, just in case you need some distraction or you are bored: https://primes.nickyreinert.de/ https://primes.nickyreinert.de/
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