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Another perhaps surprising aspect, to me at least, is the apparent uniformity of the density of prime numbers in the plane. It's my understanding that the densi

Another perhaps surprising aspect, to me at least, is the apparent uniformity of the density of prime numbers in the plane. It's my understanding that the density of the prime numbers decreases as you go higher [1], so why does the plane look so uniformly covered? [1] http://en.wikipedia.org/wiki/Prime_number_theorem http://en.wikipedia.org/wiki/Prime_number_theorem

oakwhiz 13y ago
Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.
- spikels 13y ago
  Don't think that would make any difference as long as the size of each dot remains the same. And if you look carefully you can see the density is higher in the center than around the edges (at least to my eye!). Once you get past 40 or 50 the density declines vary slowly. See chart at bottom of section 2 of this link: http://primes.utm.edu/howmany.shtml http://primes.utm.edu/howmany.shtml
yetanotherphd 13y ago
The density of primes is 1/log(n), so it does drop off, but not so fast that it would be obvious for small numbers. I tried creating giant Ulam spirals one time, and I did correct for this, which made it look a lot more uniform. I had initially hoped to find new patterns this way, but nothing turned up. While numerical experiments are fun, there are a huge number of potential avenues that could be explored. Finding the interesting ones is basically what mathematics is.
- darkmighty 13y ago
  How did you correct for it? Did you just nonlinearly scale the final picture or did you scale the axis on the spiral itself and then sampled the result?
  - yetanotherphd 13y ago
    I was producing grey scale images where each pixel represented a block of numbers. The count in each block was divided by the density of primes (1/log(n)) at the center of the block. However, this actually made the middle turn grey, because even though the mean value of each pixel was the same, the variance wasn't. So then I corrected for this by calculating a "z-score" instead. But like I mentioned before, it didn't turn up any interesting patterns.
    - darkmighty 13y ago
      Very interesting! I guess it specifically didn't turn up interesting patterns because a) you correct for density b) the distribution of primes is "noisy" (which is why they're so puzzling) and by averaging out the noise you get a fairly flat distribution
lutusp 13y ago
> why does the plane look so uniformly covered? It's a purely perceptual effect. As you track away from the center of the spiral, your eyes are following lines, which are one-dimensional, but you're tracking the lines through a plane, which is two-dimensional. The area increases rapidly as you move away from the center of the figure, but the one-dimensional lines don't have area. This gives the lines an apparent weight that they don't actually have.

oakwhiz 13y ago

Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.

spikels 13y ago
Don't think that would make any difference as long as the size of each dot remains the same. And if you look carefully you can see the density is higher in the center than around the edges (at least to my eye!). Once you get past 40 or 50 the density declines vary slowly. See chart at bottom of section 2 of this link: http://primes.utm.edu/howmany.shtml http://primes.utm.edu/howmany.shtml

yetanotherphd 13y ago

The density of primes is 1/log(n), so it does drop off, but not so fast that it would be obvious for small numbers. I tried creating giant Ulam spirals one time, and I did correct for this, which made it look a lot more uniform. I had initially hoped to find new patterns this way, but nothing turned up. While numerical experiments are fun, there are a huge number of potential avenues that could be explored. Finding the interesting ones is basically what mathematics is.

darkmighty 13y ago
How did you correct for it? Did you just nonlinearly scale the final picture or did you scale the axis on the spiral itself and then sampled the result?
- yetanotherphd 13y ago
  I was producing grey scale images where each pixel represented a block of numbers. The count in each block was divided by the density of primes (1/log(n)) at the center of the block. However, this actually made the middle turn grey, because even though the mean value of each pixel was the same, the variance wasn't. So then I corrected for this by calculating a "z-score" instead. But like I mentioned before, it didn't turn up any interesting patterns.
  - darkmighty 13y ago
    Very interesting! I guess it specifically didn't turn up interesting patterns because a) you correct for density b) the distribution of primes is "noisy" (which is why they're so puzzling) and by averaging out the noise you get a fairly flat distribution

lutusp 13y ago

> why does the plane look so uniformly covered? It's a purely perceptual effect. As you track away from the center of the spiral, your eyes are following lines, which are one-dimensional, but you're tracking the lines through a plane, which is two-dimensional. The area increases rapidly as you move away from the center of the figure, but the one-dimensional lines don't have area. This gives the lines an apparent weight that they don't actually have.