END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

I was wondering if there could be something like a 3D Newton???

I have tried;

Phoenix formula z=z^p+c*lastz



Polynomial1 formula z=z^p+z+c



Polynomial2 formula z=z^p-z+c



Manowar formula z=z^p+c+lastz



Ikenaga formula z=z^p+(c-1)*z-c



The more complex formulas do lead to more disconnected/dusty results.

More examples images here
http://softology.com.au/gallery/gallerymandelbulb.htm

Jason.

  Interesting.  I never noticed this thread or your statement until now, but I think it's going to be a LOT easier than you imagined (I was wrong... see below):  use my double complex formulas to "visualize" the mathematics.  We can easily do complex to a complex power...

  The double complex uses 2 complex number groupings to the n power, so the only change would be to have it be to the z power... simplicity in itself.  I will try and post the results in thread.  I never thought of doing that...
  Update:  not too pretty.  Discontinuous (although I only made a complex_1^complex_2 version, not complex_1^complex_2 + complex_1^n version).   In other words, looks like a major earthquake hit a normal fractal.  I'm not sure that the implementation I'm using is incorrect- it might simply be impossible too create a continuous version of this fractal <-- that might get someone working on it.   

  As to the OP, I've made the Burning Ship fractal, which is particularly promising in 3d (threads in the theory subforum 'beneath' this forum).  Extrapolating for the successful BS fractal, I realized it was simply sign management* that was the problem for z^2 Mandelbrots and corrected that (although perhaps the sign assignment method without singularities is NOT the correct one, as I've found other ones that produce nearly the amount of fractal detail).    

  *BS fractals using the absolute values of the real and imaginary components of the complex^n.

Phoenix gives interesting results. An example I made a while ago using Softology's VOC:
I called it King Mandelbulb

  Maybe.  The following might work.  I might try it out later tonight, or after the weekend.     note: some mimetex notice is popping up with the latex...  ignore it unless you are the bossman 

  We can take the real and imaginary components of .  We'd have to do it for each of the individual "complex" numbers.   Note that log refers to the natural log in all these equations!


  So we would end up with:

  The we tackle our next complex variable:

And this other part is a bit more complicated, as we need to divide out by its "complex magnitude".  It's new magnitude would be (y and iz are negative because we are dividing!):

  Note that the above latex is the magnitude of the y and z component to the -e to the [......]
Now we need to take tango to the tango:

And multiply tango by tmag:

Now we sort out our components:
new_x component= the real part of whiskey.
new y component = the imaginary part of whiskey * the real part of tango
new z component = the imaginary part of whiskey * the imaginary part of tango

  You might want to add in the regular z^n components as well... notice that the above is simply z^z. 

  Glad to see some good ideas bouncing around still!  I'll post an interesting modification for all trigonometric type fractals in a bit (I've already used it in several formulas, but it works in a general sense with cosine formulas with n=2,6,10,14...  and n=3,5,7,9.... (odd n)).

tex is not working here