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Author Topic: Another idea for 3D mandelbrot / buddhabrot  (Read 368 times)
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Alef
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« on: December 14, 2012, 03:52:10 PM »

In this site http://www.superliminal.com/fractals/index.html
There are some advanced 4D budhagramm technique, where 4D is derived from that z could be used like c.


Rotated probably antibuddhabrot

Maybe this can be simplified. So just render slices of buddhabrots starting iterating with permutation z=(-0.2;0) then z=(-0.1;0) , then z=(0;0), then z=(0.1;0), then z=(0.2;0) and put each render on another. Probably this would create mess, but  maybe one can get 3D cloud.

When you start mandelbrot set iteration with some permutation you get smaller and distorted set. This probably wount give all around bulbs, but maybe can generate something interesting.

Well, this could give another Holly Grail attempt of solid 3D mandelbrot. So render complex XY mandelbrot in slices wich have width of single pixel. Z value of pixel here can go as starting seed / permutation of mandelbrot set. Pseudocode would be as follows:
initialisation:
c=complex(Cx; Cy)
z=Cz

Iterate as complex numbers:
z=z^2+c

until:
|z|>128


Or maybe tehnique like budhagramm can give 4D mandelgramm, what could be represented in 3D. Probably not, but worth to try.

Melinda's Green site states
Quote
This page describes a natural extension to that technique into 4 dimensions. Unlike previous attempts to add new dimensions either by changing the function (especially the exponent), or by using quaternions instead of complex numbers, this new technique continues to use the unaltered function, changing only the way the object is rendered.

Rendering of Mandel/Julia set images typically begins by selecting a single initial complex point C from the image plane and then iterating Z from either 0 (Mandelbrot) or from a non-zero constant (Julia). The fundamental observation here is that the Mandelbrot/Julia set is 4-dimensional to begin with. I.E. for every combination of {Z-real, Z-imaginary, C-real, C-imaginary} (subsequently abbreviated Z.r, Z.i, C.r, C.i.), there exists a unique exit value.

Mandelbrot images are simply cross-sections of the 4D data at Z = 0, and Julia Set images are simply cross-sections at different planes. The original Buddhabrot image can be visualized as projections of the trajectories of Z values sampled from the Z = 0 plane.

This page describes my recent realization that the buddhabrot technique need not be limited to rendering just the Z trajectories beginning from a particular plane but instead can be generated from the entire 4D data set projected onto an arbitrary plane.

This extension is called a "Hologram" because any part of such an image is potentially influenced by all parts of the 4D domain.


It's Buddhagram, so if this technique would be used for 3D mandelbrot, it should be mandlegram. I thinh this could be right answer for the quest to Holly Grail. It sounds very conwincing, and if mandelbrot set is mathematical, so 3D mandelbrot set alsou should be.  It have just one problem, it's 4D not 3D, so it must be in good way projected to the 3D plane.

« Last Edit: December 17, 2012, 12:34:58 PM by Alef » Logged

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