Decomposition of exterior of Mandelbrot set

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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.

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Decomposition
• discrete
• continues

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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 L_k into 2^k+1 parts

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

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100-decomposition

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>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.

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

• First "angle doubling". "The first visible band shows the phase winding around four times times,
  the next eight, then sixteen. We conclude the phase winds around 2^n-1 times when it takes n iterations for a point to escape."
  
• Second: radial "edges ... line up over multiple bands ... and point at" the border of Mandelbrot set.
  
• As Robert P. Munafo says:
  "The Level Sets and Field Lines are superimposed, creating a sort of grid, and the "squares" of the grid are filled with N-digit binary numbers giving the first N binary digits of the external angles of field lines passing through the square. (Alternately, only the Nth binary digit is used.) Each level set is divided into 2n squares. It is easy to "read" the external arguments of points in the boundary of the Mandelbrot Set using a binary decomposition."
• rays of angles ( k / 2^n ) mod 1 can be seen

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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
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where Zy = Im(c) = imaginary part of complex number c

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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

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• N-Color Decompositions by Jules Verschueren
• Binary Decomposition by Robert P. Munafo
• tutorials by Ingvar Kullberg
• par file for fractint by Jay Hill

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

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Main page

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

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