I have stated a few times that the Mandelbrot and Julia algorithms cause the point from the pixel to be "swung and flung" throughout the map. Here "swung" is very specifically the doubling of the azimuth angle together with the squaring of its range. That is the de Moivre theorem in action. "Flung" is very specifically a translation of the point, without turning, along a vector. With the Julia
superset you have to choose that vector by selecting an X and a Y to create a
set. With Mandelbrot, the "fling vector" is chosen for you - leaving you no choice. It is the vector of the pixel under investigation, and changes throughout the image. Mandelbrot is not a superset, it is therefore just a
set.
Searching for a parameter of the output that could be used to provide topology in binocular stereoscopic fractals, I investigated
"The Last Fling" of a point, and found that the result was completely arbitrary. I created a
"pebbledash" image of where the points had landed just before being swung and flung into "darkest space" beyond the circle of radius 2.
Modifying the program allowed me to investigate the earliest locations of the points. I used this image:
This was originally just eight pixels by eight pixels in a 512 by 512 map representing X from -2 to +2 and Y from -2 to +2.
The first image I made was after a single "swing and fling". I could see that the tiny patch contained the entire image above.
Accordingly, I wrote
http://wehner.org/tools/fractals/position/pos.asm . You can save this to disk by asking to "view source". You can edit it to change the
repeats constant. I chose seven to avoid clutter. NASM, the "Net Assembler" from GNU, is also at
http://wehner.org/tools/fractals/NASM.EXE . You can click on that link and save with the asm file. Then, from the DOS prompt,
NASM POS.ASM will assemble it and
RENAME POS POS.COM gets it ready to run.
POS will then set it off, and deliver a file called
Owl.bmp. Unfortunately, Microsoft decided to force a leading capital letter on machine-made files.
The image it creates is this:
I have added the axes and numbers.
It can be seen that the original eight-by-eight group at 1 is "swung" upside-down by the de Moivre theorem, to arrive at location ESE, in the bottom right quadrant. It is also adjusted in size a little by the de Moivre range-squaring. Then it is "flung" to
2 at about 21.16 degrees to the X-axis, in the WNW direction, parallel with the vector of
1. Each pixel has a subtly different vector, however, this being Mandelbrot.
You can follow the sequence, and see how locations 3, 4, and others are reached - with the clump of pixels growing all the time.
The middle of that frame was at
-0.7265625 +0.28125. I set this as a fixed vector in
http://wehner.org/tools/fractals/julia/jman2.asm. This converted the above Mandelbrot image into a Julia image which is identical in the middle. That means that if the left is column 0 and the bottom is row 0, pixel 256,256 is identical in both images. Other pixels vary from each other, the further one goes from the middle:
The big black attractor on the right has vanished. The resemblance, however, is obvious.
Now I examined the first few "swings and flings" of this image, by means of
http://wehner.org/tools/fractals/position/jpos.asm.
The output, with axes added, is this:
It is interesting that in the third position, the image has actually shrunk before it begins to grow again.
Charles
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