complex numbersfractals
I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them.
What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines of deterministic code?
How is it possible that when zooming ever deeper and deeper, there are still completely new shapes coming up, while the algorithm remains the same?
Does the set maybe give us some deep insight into our universe or even other dimensions, as it involves complex numbers?
Based on the C code given there, Zx is just the real part $\Re z$ and Zy is just the imaginary part $\Im z$.
Remember that the complex iteration formula for the Mandelbrot set
$z=z^2+c$
when expanded to real variables looks like
$x=x^2-y^2+u$
$y=2xy+v$
where $z=x+iy$ and $c=u+iv$.
The variables with a 2 appended are the squares, as can be seen from how they were defined, so for instance $|z|=\sqrt{x^2+y^2}$ is sqrt(Zx2+Zy2) which is used for determining how much the iteration diverges (escapes to infinity), which is used by the coloring functions.
I was wondering something similar, I was trying to find a 3D mandelbrot. Not a mandelbulb, but a 3D equivilant of the mandelbrot set. When searching for it, I came across some 3D 'representations,' but they didn't really appear to be a true 3D mandelbrot set.
Then I accidentally made a 3D mandelbrot, in the program Mandelbulber. I made a hybrid fractal of the mandelbulb, and the 'general mandelbox fold' and came up with this:
The fun thing about this model is I can enter the X and Y coordinates from the traditional mandelbrot set, and it will zoom in on the same portion of the 3D model. You can even see the dendrites, and self-similar mandelbrots on the dendrites, exactly were they would be in the 2D set.
I'm not completely sure this is what you were looking for, I have been having trouble finding julias sets in the 3D model were I find them in the 2D model, but that could simply be because of the quality settings I'm using. Either way, this is the closest match to a true 3D mandelbrot I've come across. Any others have strange artifacts or artistic fluff that makes me wonder if it's just the 2D set with a lot of pretty effects.
edit:: After playing around more, this 3D mandelbrot is puzzling. With certain settings, I can see the outline of the 2D mandelbrot from a top-down view. If I use too many iterations, the spiraling branches and arms visible in the 2D set disappear and I only see smaller and smaller self-similar mandelbrots. Then if I set certain settings (mainly "DE detection") too too high of a quality on a point were you expect spirals and intricate branches, I am only shown an "oil spill". Then on other combinations of quality settings, I see wispy clouds of semi-transparent smoke over the mandelbrot set itself.
It seems most obvious that with the best of quality, the features from the 2D mandelbrot set are clouded and usually completely hidden when viewing on a microscopic scale.
I would really love to hear more input from someone (especially the question asker). I have a feeling this probably isn't what he was looking for since he appears to be somewhat of an expert in this area, and he hasn't found a 3D mandelbrot-esque figure. I only found this by chance, but it doesn't look like anything he has dismissed yet, so I'd love it my dumb luck happened to find a way to view a 'true' 3D mandelbrot.
Best Answer
Let $m(z,c) = z^2 + c$, consider the sequence of polynomials: $m(z,z),\,$ $m(m(z,z),z)$, ... , which are $$z^2+z,\quad z^4 + 2z^3 + z^2 + z,\quad z^8 + 4z^7 + 6z^6 + 6z^5 + 5z^4 + 2z^3 + z^2 + z,\quad...\quad.$$
Note that in terms of complex numbers the transformation $z \mapsto m(z,c)$ can be seen a way of twisting and squashing the sphere over itself.. if you keep kneading something you are bound to get tearing and crumpled filaments and such like.
Here are graphs of the first seven (produced by the software here):