Prime - Hilbert fractal

[Mircode]

Hi there!

I was thinking about completely new fractal ideas. One rather bizarre one that came to my mind was to go along the Hilbert curve and count the steps. I colored the current pixel white if its index is prime and black if it's not.

I didn't expect much from this, since it's completely far-fetched. It would probably just look like noise, but my curiosity was stronger.
The more I was suprised by the results I got.

8x8 and 16x16 and so on didn't show anything interesting. But for higher recursion levels, there seemed to appear large areas without a prime number in them. They are even connected over huge gaps where two totally different numbers are neighboring. The shapes look pretty interesting.

Here is a level 12 pic, 4096x4096px in size:
http://www.stud.tu-ilmenau.de/~miku3933/WebSpace/mathstuff/primehilbert12.html

Can anyone think of a connection between prime numbers and the Hilbert curve? Maybe the Sieve of Eratosthenes has something todo with it.

Anyway, I think it is an interesting effect and I hope it has not been discovered so far!

Greetings,
  Mirko

[reesej2]

Oh, what a fascinating phenomenon! Positively spectacular! 

[cKleinhuis]

that is the spirit ... surprisingly, i had similar results, when counting the bulbs of a mandelbulb, and dotted them on my wall ...

[Timeroot]

Heheheh... since you linked it in the form of an HTML - instead of the raw picture - I had a hard time reading it. But as soon as I saw how it shot out, without even distinguishing a letter, I could guess what it was going to be. Very nice!

It would be fun to try actually doing that, with the primes and the Hilbert curve... it could also be tried with other space-filling curves.

This reminds me of one xkcd where the internet's 256 sections of IP addresses are plotted along a hilbert curve. Kinda fun to see "Europe's internet" right there, "America's internet" over here... Good chucklings!

[Mircode]

Heheheh... since you linked it in the form of an HTML - instead of the raw picture - I had a hard time reading it.

Well, that was the whole point. It should still look a little real when just the upper left corner was displayed
And firerfox would have scaled it so it fits on screen if I had just linked the picture.

It would be fun to try actually doing that

The background actually is that. I really tried it some days ago. For this april pic I just multiplied the result with a mask so that the letters would appear. It starts at the upper left corner, you can see kind of gradient contrast lines there due to the previously mentioned gaps and because the density of prime numbers decreases with higher numbers. Maybe try to shrink the pic with a good algorithm, it becomes more visible.

This reminds me of one xkcd where the internet's 256 sections of IP addresses are plotted along a hilbert curve.

This is where the idea for this "fractal" was born