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/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 ?
January 28, 2015, 08:12:30 AM
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