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In writing a class library for some of the common classes I use in my fractal programs something occurred to me.

This is the polynomial I would like to discuss. I hope my notation is correct. I would hate to make a fool of my self again today.

(http://i126.photobucket.com/albums/p114/chaos_5/equation.png)

I was writing the code for a cubic Mandelbrot and discovered that I can derive a standard Mandelbrot, a Julia, a cubic Mandelbrot, and many other variations of these classic fractals by simply using different variables for i, C, a0, a1, a2 etc.

For example:
Mandelbrot i=2, a2 = 1, a1 = 1, a0 = 0, C=0
Julia i=2, a2 = 1, a1 =0, a0=0, C= -1+0i
Cubic Mandelbrot i=3 a3=1, a2 = 0, a1 = 1, a0 = 0, C=0

Im sure you all get the idea.
Ive found a lot of interesting variations using different values as well.

I do have one question I was hoping someone could answer.
Is there a name for this general expression?

minor thing: considering z^0 == 1, a_0 would be what you currently have as C. in other words, the following suffices to represent the whole series:
(http://www.fractographer.com/propaganda/polysum.png)

the expression on the rhs is just a (general) polynomial, so i guess the whole thing is just a polynomial equation.

while thinking about that expression it occurred to me that the a_k needn't be real; the formulation of the equation reminds one of a fourier series, so why not replace it with one and see what happens when it's iterated?

(http://www.fractographer.com/propaganda/harmonicpoly.png)

in this expression, several harmonics are summed together to form the complex w_k; the harmonic frequencies are given by f_h, which should be made to have some common factor (otherwise they won't be harmonics!).

now i haven't tried this so i can't guarantee that it'll produce nice results, but it seems like a viable avenue of exploration:

how many harmonics to use?
which frequencies to use?
should they have a common multiple, or should the frequencies be primes?
should the f_h == 0  case be included? it will need a normalisation factor of 1/mn to produce z_n+1 = z_n.
how many terms to use? what happens with more vs fewer terms?
etc...

I did wonder what to do with the Z^0 since it is always 1. I thought that perhaps starting the series at i=1 would be a good idea.

As for C, I have tryed using complex numbers with good results.

Thank you for the feedback!

eek, the alphablending doesn't seem to be working :| anyway, i didn't mean just making C complex, i meant all the a-coefficients; moreover, C is completely unnecessary, a_0 gives the constant value (independent of z).

once i've worked on my gui a bit more i'll investigate the harmonic series i posted.