Could the 2D equation Z=z^+c (Mandelbrot set) be cubed to create 3D fractal images?

Z=z3rd+c Z=z4th+c Z=z5th+c Z=znth+c etc etc etc

Could the 2D equation Z=z^+c (Mandelbrot set) be cubed to create 3D fractal images?
No. Iterating the equation Z(n) = Z(n-1)^x + c will always return complex numbers. (Remember that the Mandelbrot Set is an image graphed on the complex plane.)





However, you do get interesting images if you use integer powers greater than two. (You also get interesting pictures if you use integer powers that are _less_ than -1.


A nice website that has these images can be found at http://yu.jason.googlepages.com/themande... )








Now -- if you _really_ want 3D images that are related to the Mandelbrot Set, you could use quaternions (which are 4th dimensional values), and try to visualize their shadows in 3-space. (The website at http://www.ibiblio.org/e-notes/MSet/Quat... can give you an introduction to this, and the website at http://www.superliminal.com/fractals/ can give you some images to get you started.)





Or you can just make textured, 3-dimensional-looking pictures of the 2-dimensional Mandelbrot Set. The website at http://www.mandelbrot-dazibao.com/Bump/B... has images like this.








Hope this helps!





:-)