5 ms·

I like to put things on x/y-charts to get a visual overview. Even things that usually are not displayed in this fashion. When it comes to prime numbers, it alw

by mg 4y ago

I like to put things on x/y-charts to get a visual overview. Even things that usually are not displayed in this fashion. When it comes to prime numbers, it always irked me, that one cannot easily look at them in a visual way. I always thought that they would probably look beautiful. One day, I had the idea to counter this by visualizing prime numbers in the complex plane. I then wrote a little script to colorize each number in the complex plane by how "prime" it is. And indeed, the result looks beautiful: https://www.gibney.org/does_anybody_know_this_fractal https://www.gibney.org/does_anybody_know_this_fractal Then I went back to do more mundane things with x/y charts. The one that became most popular is Product Chart: https://www.productchart.com https://www.productchart.com But every once in a while, I still dream about the "complex divisor fractal" and what mysteries might be hidden inside of it.

CapmCrackaWaka 4y ago
I have seen many “plots” of prime numbers that look really good. In addition to looking good, I sometimes _swear_ I see a pattern. It’s really frustrating, to see something on a graph which looks like it follows some pattern or algorithm[1], and to know that finding the mathematical definition of that pattern would would be revolutionary, world news. I can see why so many people get sucked into this problem. [1] https://www.reddit.com/r/math/comments/7v02i6/comparison_between_5000_and_50000_prime_numbers/ https://www.reddit.com/r/math/comments/7v02i6/comparison_bet...
- CyborgCabbage 4y ago
  Knew I remembered that spiral from somewhere https://www.youtube.com/watch?v=EK32jo7i5LQ https://www.youtube.com/watch?v=EK32jo7i5LQ
  - hinkley 4y ago
    This is a very good video. The tl;dr on it is that all primes are of the pattern 3n±1, and 2𝞹 (polar coordinates) is .283 mod 3, hence the primary curve. Then he explains about fractions that approximate pi, but I think he's over-explaining. Reducing a fraction means making the denominator and numerator co-prime, and in the case of pi approximations, the interesting ones all have prime numbers as the denominator, so the 3n±1 pattern appears again.
- mg 4y ago
  There is a movie from 1998 called "pi" about a number theorist who got sucked into it: https://en.wikipedia.org/wiki/Pi_(film) https://en.wikipedia.org/wiki/Pi_(film) I saw it a long time ago, but still vividly remember some of the scenes.
  - kortilla 4y ago
    Yeah, some of that movie really gets drilled into your head.
mananaysiempre 4y ago
You have seen the Ulam spiral[1] from 1963, right? I don’t know that it provides any actual insight into this stupenduously complicated[2] topic, but if you want something to draw about prime numbers that’s probably the best-known option. Or see the story about “primons”[3] for a view that’s less visual but no less tangible (given a certain kind of background at least). And yes, as you note, your CDF very much looks like a square grid after an inversion. That is to say: draw a square grid on a horizontal plane, lay down a ball on that plane, project the image onto the surface of the ball by straight rays through its topmost point (imagine a lightbulb there; see? “projection”), then take a parallel plane touching that same ball and project back onto it from the surface (now casting rays from the bottommost point). But if so, it’s boring in that it should have little to do with prime numbers. Let me think about this. (The story of complex numbers and projective transformations is not boring, of course, it’s quite pretty, just doesn’t provide much of an insight into the primes; and, in turn, its connection to hyperbolic geometry, most widely known through Escher’s drawings, is also quite wonderful, but removed even further from the original picture, so might not be necessary to understand it.) [1] https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral [2] https://arxiv.org/abs/math/0210327 https://arxiv.org/abs/math/0210327 [3] https://math.ucr.edu/home/baez/week199.html https://math.ucr.edu/home/baez/week199.html
- shaunxcode 4y ago
  The thing I found was that if you start the spiral with a 0 you get a similar pattern. Further if you layer each of the triangular sections you get a really cool pattern[0]. [0] https://lh4.googleusercontent.com/proxy/Ls_8QQgRDf4oK6FwliJ7vQYFrmB964h3mcmNAQEkVOxBIhXQn8Wq_gOrtU1bI_ZlRMQjfW9fV7z-YzBUnWWPvXj2hTD2wHfMwRa-3Xhy_uleeDMBNJwTs7nRZDp2pkkn https://lh4.googleusercontent.com/proxy/Ls_8QQgRDf4oK6FwliJ7...
noasaservice 4y ago
Hmm.. I keep looking at your first ( https://www.gibney.org/images/size02/-1_-1_to_1_1.jpg https://www.gibney.org/images/size02/-1_-1_to_1_1.jpg ) image, and I swear it looks 3 dimensional positively curved space, with vertices intersecting in depth. Just that since it's a picture, its being compressed from 3d->2d. Even though the lines are curved, I'm parsing them as straight.. somehow. Perhaps this is the wrong projection, and maybe it needs to be mapped on a sphere using lat/lon/alt (parametric)? That would also explain the infinite border asymptotes, and going from 2d mercator->spherical would modify those.
pilotneko 4y ago
Nice work. You might be interested in this visualization of the first million prime numbers from UMAP: https://umap-learn.readthedocs.io/en/latest/exploratory_analysis.html https://umap-learn.readthedocs.io/en/latest/exploratory_anal...
coldtea 4y ago
>When it comes to prime numbers, it always irked me, that one cannot easily look at them in a visual way. I always thought that they would probably look beautiful. There are several of visualizations of prime numbers, and they do, look beautiful. E.g.: https://jaketae.github.io/study/prime-spirals/ https://jaketae.github.io/study/prime-spirals/ https://www.cantorsparadise.com/unexpected-beauty-in-primes-b347fe0511b2 https://www.cantorsparadise.com/unexpected-beauty-in-primes-... https://mathworld.wolfram.com/PrimeSpiral.html https://mathworld.wolfram.com/PrimeSpiral.html
frogperson 4y ago
There are many visualizations of the primes in 2D and many are fascinating. I've always wondered what it would look like in 3d or 4d or higher. what if the primes were mapped on to some crazy topological shape? Is there some shape and dimension out there that produces a "perfect" pattern?
- q-big 4y ago
  > I've always wondered what it would look like [...] if the primes were mapped on to some crazy topological shape? This is easy: consider the spectrum of the ring Z, Spec Z (by definition, the spectrum of a commutative ring consists of its prime ideals; all prime ideals of Z (with the exception of the zero ideal) are generated by the prime numbers; in this sense, we can identify all elements of Spec Z (with the exception of the zero ideal, which is "just another point" of Spec Z) with the prime numbers if we desire). On Spec Z, we can define a topology, the Zariski topology, by defining V(I) := {P \in Spec Z | I \subseteq P}, where I is an ideal of Z, as the closed sets of the Zariski topology. See * https://en.wikipedia.org/w/index.php?title=Zariski_topology&oldid=1091588561#Spectrum_of_a_ring https://en.wikipedia.org/w/index.php?title=Zariski_topology&... * https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_ring&oldid=1076963164#Zariski_topology https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_rin... for details.
  - voldacar 4y ago
    I'm not sure that this answers the parent's question. The Zariski topology is unusual, and in "some crazy topological shape" I think the parent is looking for a way to map the primes onto a smooth manifold in some finite dimensional space in a way that elucidates some kind of hidden structure.
akomtu 4y ago
That prime fractal seems a good fit for shadertoy: it would give GPU perf and basic zoom in zoom out for free. The last two images look indeed suspiciously good.
deleted 4y ago
[deleted]

by mg 4y ago

CapmCrackaWaka 4y ago

I have seen many “plots” of prime numbers that look really good. In addition to looking good, I sometimes _swear_ I see a pattern. It’s really frustrating, to see something on a graph which looks like it follows some pattern or algorithm[1], and to know that finding the mathematical definition of that pattern would would be revolutionary, world news. I can see why so many people get sucked into this problem. [1] https://www.reddit.com/r/math/comments/7v02i6/comparison_between_5000_and_50000_prime_numbers/ https://www.reddit.com/r/math/comments/7v02i6/comparison_bet...

CyborgCabbage 4y ago
Knew I remembered that spiral from somewhere https://www.youtube.com/watch?v=EK32jo7i5LQ https://www.youtube.com/watch?v=EK32jo7i5LQ
- hinkley 4y ago
  This is a very good video. The tl;dr on it is that all primes are of the pattern 3n±1, and 2𝞹 (polar coordinates) is .283 mod 3, hence the primary curve. Then he explains about fractions that approximate pi, but I think he's over-explaining. Reducing a fraction means making the denominator and numerator co-prime, and in the case of pi approximations, the interesting ones all have prime numbers as the denominator, so the 3n±1 pattern appears again.
mg 4y ago
There is a movie from 1998 called "pi" about a number theorist who got sucked into it: https://en.wikipedia.org/wiki/Pi_(film) https://en.wikipedia.org/wiki/Pi_(film) I saw it a long time ago, but still vividly remember some of the scenes.
- kortilla 4y ago
  Yeah, some of that movie really gets drilled into your head.

mananaysiempre 4y ago

You have seen the Ulam spiral[1] from 1963, right? I don’t know that it provides any actual insight into this stupenduously complicated[2] topic, but if you want something to draw about prime numbers that’s probably the best-known option. Or see the story about “primons”[3] for a view that’s less visual but no less tangible (given a certain kind of background at least). And yes, as you note, your CDF very much looks like a square grid after an inversion. That is to say: draw a square grid on a horizontal plane, lay down a ball on that plane, project the image onto the surface of the ball by straight rays through its topmost point (imagine a lightbulb there; see? “projection”), then take a parallel plane touching that same ball and project back onto it from the surface (now casting rays from the bottommost point). But if so, it’s boring in that it should have little to do with prime numbers. Let me think about this. (The story of complex numbers and projective transformations is not boring, of course, it’s quite pretty, just doesn’t provide much of an insight into the primes; and, in turn, its connection to hyperbolic geometry, most widely known through Escher’s drawings, is also quite wonderful, but removed even further from the original picture, so might not be necessary to understand it.) [1] https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral [2] https://arxiv.org/abs/math/0210327 https://arxiv.org/abs/math/0210327 [3] https://math.ucr.edu/home/baez/week199.html https://math.ucr.edu/home/baez/week199.html

shaunxcode 4y ago
The thing I found was that if you start the spiral with a 0 you get a similar pattern. Further if you layer each of the triangular sections you get a really cool pattern[0]. [0] https://lh4.googleusercontent.com/proxy/Ls_8QQgRDf4oK6FwliJ7vQYFrmB964h3mcmNAQEkVOxBIhXQn8Wq_gOrtU1bI_ZlRMQjfW9fV7z-YzBUnWWPvXj2hTD2wHfMwRa-3Xhy_uleeDMBNJwTs7nRZDp2pkkn https://lh4.googleusercontent.com/proxy/Ls_8QQgRDf4oK6FwliJ7...

noasaservice 4y ago

Hmm.. I keep looking at your first ( https://www.gibney.org/images/size02/-1_-1_to_1_1.jpg https://www.gibney.org/images/size02/-1_-1_to_1_1.jpg ) image, and I swear it looks 3 dimensional positively curved space, with vertices intersecting in depth. Just that since it's a picture, its being compressed from 3d->2d. Even though the lines are curved, I'm parsing them as straight.. somehow. Perhaps this is the wrong projection, and maybe it needs to be mapped on a sphere using lat/lon/alt (parametric)? That would also explain the infinite border asymptotes, and going from 2d mercator->spherical would modify those.

pilotneko 4y ago

Nice work. You might be interested in this visualization of the first million prime numbers from UMAP: https://umap-learn.readthedocs.io/en/latest/exploratory_analysis.html https://umap-learn.readthedocs.io/en/latest/exploratory_anal...

coldtea 4y ago

>When it comes to prime numbers, it always irked me, that one cannot easily look at them in a visual way. I always thought that they would probably look beautiful. There are several of visualizations of prime numbers, and they do, look beautiful. E.g.: https://jaketae.github.io/study/prime-spirals/ https://jaketae.github.io/study/prime-spirals/ https://www.cantorsparadise.com/unexpected-beauty-in-primes-b347fe0511b2 https://www.cantorsparadise.com/unexpected-beauty-in-primes-... https://mathworld.wolfram.com/PrimeSpiral.html https://mathworld.wolfram.com/PrimeSpiral.html

frogperson 4y ago

There are many visualizations of the primes in 2D and many are fascinating. I've always wondered what it would look like in 3d or 4d or higher. what if the primes were mapped on to some crazy topological shape? Is there some shape and dimension out there that produces a "perfect" pattern?

q-big 4y ago
> I've always wondered what it would look like [...] if the primes were mapped on to some crazy topological shape? This is easy: consider the spectrum of the ring Z, Spec Z (by definition, the spectrum of a commutative ring consists of its prime ideals; all prime ideals of Z (with the exception of the zero ideal) are generated by the prime numbers; in this sense, we can identify all elements of Spec Z (with the exception of the zero ideal, which is "just another point" of Spec Z) with the prime numbers if we desire). On Spec Z, we can define a topology, the Zariski topology, by defining V(I) := {P \in Spec Z | I \subseteq P}, where I is an ideal of Z, as the closed sets of the Zariski topology. See * https://en.wikipedia.org/w/index.php?title=Zariski_topology&oldid=1091588561#Spectrum_of_a_ring https://en.wikipedia.org/w/index.php?title=Zariski_topology&... * https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_ring&oldid=1076963164#Zariski_topology https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_rin... for details.
- voldacar 4y ago
  I'm not sure that this answers the parent's question. The Zariski topology is unusual, and in "some crazy topological shape" I think the parent is looking for a way to map the primes onto a smooth manifold in some finite dimensional space in a way that elucidates some kind of hidden structure.

akomtu 4y ago

That prime fractal seems a good fit for shadertoy: it would give GPU perf and basic zoom in zoom out for free. The last two images look indeed suspiciously good.

deleted 4y ago

[deleted]