Decomposition of exterior of Mandelbrot set





Decomposition is modification of escape time algorithm.
The target set T = { z : |zn| > R } with a very large escape radius ( I use R = 12 )
is divided into : (2 or more ) parts.


Binary decomposition = 2-decomposition
Target set is divided into 2 parts.

Let's choose exterior of circle ( center = 0; radius = escape radius = bailout) as a target set.
The circle target set we divide into:
-upper half ( white) T u = { z : |zn| > R and Im (zn) > 0}
- lower half (black) T l = { z : |zn| > R and Im (zn) < = 0 }


Division of target set induces division of level Lk into 2k+1 parts


8-decomposition
compare with Figure 10.1 made by Professor Douglas C. Ravenel


100-decomposition


>100-decomposition = continues decomposition
This picture shows the phase (angle) of the last point of orbit. Compare Linas version

One can see that borders of bands ( contour lines ) do not meet at point (c = -2). It means that for escape radius > 2.

Let's explore these images with Linas.
We can see a few things.

How to speed up calculations of binary decomposition by checking not agle = Arg(Zn) but sign of Im(Zn).
See : An open letter to Dr. Meech from Joyce Haslam in FRACTAL REPORT 27 
Thx for Roger Bagula for this information
"Binary decomposition is a nice and easy black & white ... display.
Plot the interior of the set in either white or a colour (I usually choose yellow).
Outside the set, plot the point in black if the final value of y is positive,
but in white if the final value of y is negative."

So we can write drawing procedure in pseudocode:

if (iteration=IterationMax)
then color:=Black // Mandelbrot set
else if Zy > 0
  then color:=White // upper half
  else color:=Black; // lower half
---------------------
where Zy = Im(c) = imaginary part of complex number c

Another type of decomposition ( here one does not see level sets ), so external rays should be better seen:

On above image external rays of angle: 1/16, 2/16, .. , 16/16 should be seen.
( I'm not quite sure that is correct image ).
And here is correct image made with Mandel - program for DOS by Wolf Jung, version 3.5 here one can see rays of the form n / 16




All above images are made with my program Mandelbrot set explorer.


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Autor: Adam Majewski
adammaj1-at-o2-dot-pl
Feel free to e-mail me. (:-))

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