ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Thursday, January 14, 2010

Self similarity of z/log(z)

I made the following pics by temporally iterating 3 spatial iterations;

z/log(z)
(z/log(z)) / log(z/log(z))
[(z/log(z)) / log(z/log(z))] / log[(z/log(z)) / log(z/log(z))]

The temporal iteration number is 170. Bailout is 40. The 4th pic is a close-up of the 3rd. They came about after a Google group discussion.

The 3 pictures similarity is amazing I thought. Just like the temporal iteration increasing resolution, the spatial iteration seems to do the same. I think there is a strong link here to the prime numbers, whose distribution is known to be self similar. Apart from the known fact that the primes are semi uniformly distributed in x/log(x), they seem to obey the H-fractal in their further patterns.

Another very interesting point here is that the both temporal and spatial iterations limit to 'e' at the outer region of the shape;

...(z/log(z)) / log(z/log(z))... -> e

demonstrating the binary structure of 'e'.









































































The tip of the last amplified to the limit of my pc. Note the slight angle. Numbers seem to prefer a side, as far as this function is concerned. This may be due to the accumulation of errors. If not, then this is also very interesting.



















The semi-stable points. What is the link to the zeros of the Riemann's Zeta function?

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