Title: [Buddhabrot] Choosing randomly on the mercator / log map
Post by: tit_toinou on January 28, 2014, 01:02:57 PM
Hi !
Again with buddhabrot, has anyone ever tried to choose randomly points in the mercator map around an interesting point (a minibrot for example) of the mandelbrot ?
The mercator map is the complex log map i.e. for instead of iterating z -> z^2 + c you are iterating z -> z^2 + exp(c) + a where a is an interesting point.
Then you are just looking at : imaginary part of c between 0 and TWOPI, real part of c inferior of 2 (because of the bailout). And if real part of c decrease you are approaching the interesting point.
Examples of mercator map : http://www.flickr.com/photos/arenamontanus/sets/72157615740829949/ (http://www.flickr.com/photos/arenamontanus/sets/72157615740829949/)
Here instead of doing)
, )
(for example) you are doing  ))
, )
in polar coordinates around a :  + a_{x})
, + a_{y})
.
This means that you have more chance to pick a point nearer the interesting point.
First discovery : this technique is good for finding high iteration points of Mandelbrot. When doing "not fully developed" buddhabrot with this technique I get buddhabrot faster (using float precision and something like iterations are between 100k and 500k).
Second discovery : Zooming with UltraFractal to a minibrot, taking that location as the interesting point "a", lauching my render, I get fully developed Buddhabrot.... that looks the same as regular Buddhabrot !
It means that "everywhere where there's interesting features of mandelbrot", the trace of theses points' iterations on the complex plane always look the same.
Is there only ONE "fully developed" Buddhabrot ?
Again with buddhabrot, has anyone ever tried to choose randomly points in the mercator map around an interesting point (a minibrot for example) of the mandelbrot ?
The mercator map is the complex log map i.e. for instead of iterating z -> z^2 + c you are iterating z -> z^2 + exp(c) + a where a is an interesting point.
Then you are just looking at : imaginary part of c between 0 and TWOPI, real part of c inferior of 2 (because of the bailout). And if real part of c decrease you are approaching the interesting point.
Examples of mercator map : http://www.flickr.com/photos/arenamontanus/sets/72157615740829949/ (http://www.flickr.com/photos/arenamontanus/sets/72157615740829949/)
Here instead of doing
This means that you have more chance to pick a point nearer the interesting point.
First discovery : this technique is good for finding high iteration points of Mandelbrot. When doing "not fully developed" buddhabrot with this technique I get buddhabrot faster (using float precision and something like iterations are between 100k and 500k).
Second discovery : Zooming with UltraFractal to a minibrot, taking that location as the interesting point "a", lauching my render, I get fully developed Buddhabrot.... that looks the same as regular Buddhabrot !
It means that "everywhere where there's interesting features of mandelbrot", the trace of theses points' iterations on the complex plane always look the same.
Is there only ONE "fully developed" Buddhabrot ?