6 ms·

this chart basically includes a set of periodic curves, even though it looks nice but what does it tell you about prime numbers ? Intersected by only two curves

by thlt 14y ago

this chart basically includes a set of periodic curves, even though it looks nice but what does it tell you about prime numbers ? Intersected by only two curves 1 and itself ??? well, everyone knows this, no need to make a chart.

nessus42 14y ago
It tells you that prime numbers and periodic curves might be related more than you might have thought at first blush. I.e., if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. As I mentioned, I don't know much about number theory, but I do know that it uses a lot of math that is counterintuitive at first blush. Perhaps this visualization gives us a clue as to why that is the case.
- dxbydt 14y ago
  >if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. Dude, no offense, you are really making stuff up. The periodicity has to to with the divisors of the composites. Every prime p has exactly 2 curves - the wave of period 1, and the wave of period p. There's nothing interesting or useful to take way from that observation. otoh, you look at a composite c - it has several divisors & each divisor d generates a curve of period d, and those curves intersect in interesting ways...though I don't see how you could mathematically analyze them to tell you anything about the primes nearby. They are mostly pretty patterns, not mathematically useful...here are 2 quite famous & useful diagrams on periodicity in primes if you are interested in that sort of thing - 1. the prime number cross - http://img841.imageshack.us/img841/1329/primenumbercross.gif http://img841.imageshack.us/img841/1329/primenumbercross.gif 2. hippocampal neurons & primes - http://www.hindawi.com/journals/amp/2011/519178/fig8/ http://www.hindawi.com/journals/amp/2011/519178/fig8/
  - nessus42 14y ago
    I take it, then, that you find nothing inspirational in the videos made by Vi Hart either. To each, their own. As to the diagrams you pointed me at, they mean nothing to me, and do not inspire me. If I knew more about number theory, perhaps they would. Some things are not about information, they are about inspiration. The visualization in the OP provides inspiration that number theory is connected to other fields of math. You can see structure in the way that the periodic curves intersect and the primes are the gaps. If you could understand that structure, then maybe you could understand the gaps. Then again maybe not. That doesn't mean that the question and the visualization doesn't cause you to think and wonder. Sure, this is old hat to mathematicians, and for all I know, this approach is a complete dead end. Sometimes dead ends are interesting too.
  - Retric 14y ago
    It does give you a good intuition as to why twin primes aka N, N+2 are so common.
    - thlt 14y ago
      how so ?
      - Retric 14y ago
        The number between them have a high number of small factors including 2 and 3. 2 * 3 = 6: 5 and 7 are prime. 2 * 3 * 2 = 12: 11 and 13 are prime 2 * 3 * 3 = 18: 17 and 19 are prime 2 * 3 * 5 = 30: 11 and 13 are prime 2 * 3 * 7 = 42: 41 and 43 are prime. However, 2 * 3 * 101 = 606 but 605 is not prime. But, 2 * 3 * 5 * 5 = 150 and 149 and 151 are prime. 2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.
        jjaredsimpson 14y ago
        I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least. Smaller numbers have multiples that are more densely distributed among the integers.
        Retric 14y ago
        The point is twin primes (6k-1, 6k+1) are more likely for a large k when k is a composite number than a prime AND the more factors of k the higher chance for twin primes. EX: K = (6 * 2 * 3 * 5 * 7 * 11 * 13 ) gives a twin prime.
  - yaks_hairbrush 14y ago
    > Dude, no offense, you are really making stuff up. No. Parent has it essentially correct. Many new results in number theory are obtained by studying automorphic forms, which are the stable waveforms, on various spaces. Things like the Riemann zeta function arise out of spectral transforms of automorphic forms.
    - nessus42 14y ago
      Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. The visualization in the OP shows how to use sine waves to build a sieve of Sieve of Eratosthenes. Now that I've seen the visualization, this revelation seems so utterly obvious that it goes without saying. But somehow, I never drew this connection until seeing the visualization. And once I see how "obvious" this is, it's suddenly obvious how e and pi might get into everything, because everything that repeats with a specific frequency can be modeled as a wheel rolling along and leaving a mark on every revolution. And what is multiplication, but repeated addition? I.e., a wheel of a certain size rolling down the number line, leaving its mark once per turn. Above a certain age, we tend to stop thinking about multiplication as repeated addition, and so we don't think about how all multiplication is implicitly bringing pi into everything we are doing. Maybe everything I said above is wrong in some way, since, as I have mentioned, I haven't studied any math past calculus and college algebra, and even that was so long ago, most of it I don't remember. Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" But there must be some way to interpret what I just wrote that doesn't deserve being summarily shot down.
      - yaks_hairbrush 14y ago
        You're welcome! > One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. Yep. Sure blew my mind when I saw a proof of quadratic reciprocity (a very neat result about square numbers in modular arithmetic) which used complex analysis (how on earth can complex numbers prove stuff about modular arithmetic!?) > Maybe everything I said above is wrong in some way... Not really. Your intuition upon seeing this visualization was pretty much right on: studying periodic functions is a way of understanding numbers. > Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" Not so much. It took some mighty smart folks to develop some ideas which are perhaps suggested, in hindsight, by this picture. The big one is Fourier series and transforms, which allow you to decompose periodic functions into their constituent sine waves. You can use Fourier analysis to get information about number theory, which was essentially your suggestion. However, that's not at all obvious without seeing this picture. Certainly, my first exposures to Fourier analysis were in the context of signal processing and solving PDEs. I had absolutely no inkling that it may be useful for number theory until actually seeing it. Even if I had seen this picture 7 years ago (when I knew signal processing and PDEs, but not number-theoretic applications), I probably would not have made the connection that you made. So, I think your intuition was a rather non-obvious idea, and so your comment did not deserve the quick shoot-down. (And even if it were obvious to folks who have studied math, it would still be non-obvious to someone, probably).

by thlt 14y ago

nessus42 14y ago

It tells you that prime numbers and periodic curves might be related more than you might have thought at first blush. I.e., if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. As I mentioned, I don't know much about number theory, but I do know that it uses a lot of math that is counterintuitive at first blush. Perhaps this visualization gives us a clue as to why that is the case.

dxbydt 14y ago
>if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. Dude, no offense, you are really making stuff up. The periodicity has to to with the divisors of the composites. Every prime p has exactly 2 curves - the wave of period 1, and the wave of period p. There's nothing interesting or useful to take way from that observation. otoh, you look at a composite c - it has several divisors & each divisor d generates a curve of period d, and those curves intersect in interesting ways...though I don't see how you could mathematically analyze them to tell you anything about the primes nearby. They are mostly pretty patterns, not mathematically useful...here are 2 quite famous & useful diagrams on periodicity in primes if you are interested in that sort of thing - 1. the prime number cross - http://img841.imageshack.us/img841/1329/primenumbercross.gif http://img841.imageshack.us/img841/1329/primenumbercross.gif 2. hippocampal neurons & primes - http://www.hindawi.com/journals/amp/2011/519178/fig8/ http://www.hindawi.com/journals/amp/2011/519178/fig8/
- nessus42 14y ago
  I take it, then, that you find nothing inspirational in the videos made by Vi Hart either. To each, their own. As to the diagrams you pointed me at, they mean nothing to me, and do not inspire me. If I knew more about number theory, perhaps they would. Some things are not about information, they are about inspiration. The visualization in the OP provides inspiration that number theory is connected to other fields of math. You can see structure in the way that the periodic curves intersect and the primes are the gaps. If you could understand that structure, then maybe you could understand the gaps. Then again maybe not. That doesn't mean that the question and the visualization doesn't cause you to think and wonder. Sure, this is old hat to mathematicians, and for all I know, this approach is a complete dead end. Sometimes dead ends are interesting too.
- Retric 14y ago
  It does give you a good intuition as to why twin primes aka N, N+2 are so common.
  - thlt 14y ago
    how so ?
    - Retric 14y ago
      The number between them have a high number of small factors including 2 and 3. 2 * 3 = 6: 5 and 7 are prime. 2 * 3 * 2 = 12: 11 and 13 are prime 2 * 3 * 3 = 18: 17 and 19 are prime 2 * 3 * 5 = 30: 11 and 13 are prime 2 * 3 * 7 = 42: 41 and 43 are prime. However, 2 * 3 * 101 = 606 but 605 is not prime. But, 2 * 3 * 5 * 5 = 150 and 149 and 151 are prime. 2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.
      - jjaredsimpson 14y ago
        I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least. Smaller numbers have multiples that are more densely distributed among the integers.
        Retric 14y ago
        The point is twin primes (6k-1, 6k+1) are more likely for a large k when k is a composite number than a prime AND the more factors of k the higher chance for twin primes. EX: K = (6 * 2 * 3 * 5 * 7 * 11 * 13 ) gives a twin prime.
- yaks_hairbrush 14y ago
  > Dude, no offense, you are really making stuff up. No. Parent has it essentially correct. Many new results in number theory are obtained by studying automorphic forms, which are the stable waveforms, on various spaces. Things like the Riemann zeta function arise out of spectral transforms of automorphic forms.
  - nessus42 14y ago
    Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. The visualization in the OP shows how to use sine waves to build a sieve of Sieve of Eratosthenes. Now that I've seen the visualization, this revelation seems so utterly obvious that it goes without saying. But somehow, I never drew this connection until seeing the visualization. And once I see how "obvious" this is, it's suddenly obvious how e and pi might get into everything, because everything that repeats with a specific frequency can be modeled as a wheel rolling along and leaving a mark on every revolution. And what is multiplication, but repeated addition? I.e., a wheel of a certain size rolling down the number line, leaving its mark once per turn. Above a certain age, we tend to stop thinking about multiplication as repeated addition, and so we don't think about how all multiplication is implicitly bringing pi into everything we are doing. Maybe everything I said above is wrong in some way, since, as I have mentioned, I haven't studied any math past calculus and college algebra, and even that was so long ago, most of it I don't remember. Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" But there must be some way to interpret what I just wrote that doesn't deserve being summarily shot down.
    - yaks_hairbrush 14y ago
      You're welcome! > One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. Yep. Sure blew my mind when I saw a proof of quadratic reciprocity (a very neat result about square numbers in modular arithmetic) which used complex analysis (how on earth can complex numbers prove stuff about modular arithmetic!?) > Maybe everything I said above is wrong in some way... Not really. Your intuition upon seeing this visualization was pretty much right on: studying periodic functions is a way of understanding numbers. > Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" Not so much. It took some mighty smart folks to develop some ideas which are perhaps suggested, in hindsight, by this picture. The big one is Fourier series and transforms, which allow you to decompose periodic functions into their constituent sine waves. You can use Fourier analysis to get information about number theory, which was essentially your suggestion. However, that's not at all obvious without seeing this picture. Certainly, my first exposures to Fourier analysis were in the context of signal processing and solving PDEs. I had absolutely no inkling that it may be useful for number theory until actually seeing it. Even if I had seen this picture 7 years ago (when I knew signal processing and PDEs, but not number-theoretic applications), I probably would not have made the connection that you made. So, I think your intuition was a rather non-obvious idea, and so your comment did not deserve the quick shoot-down. (And even if it were obvious to folks who have studied math, it would still be non-obvious to someone, probably).