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I pretty much think that the search for the perfect 3d mandelbrot is over. My vote goes to the Nylander twinbee formula.

But having gone through the experience with you all i saw the vast array of mandelbrot 3d forms in my search. Every now and again i look at a new formulation and test it against the nylander form. The images are weird and wonderful and can be instructive.

I hope that this might be a thread for people like myself and fracmonk and Benesi etc to share half baked ideas or fully formed ones with others who have the inclination to explore them.

Here is a mandy formula to start it off with

q=r^n*(exp(i*n*ø)+cos(n*Ω))*j+c

Where ø and Ω are radian measures of angle and n is the signal/ log of the polynomial numeral. Q is the quad form as c is a quad value

but it can easily be written in the "complex/triplex" form z with c#

For initial exploration i used x#,y#,imaj(z) for radian measures  and n=2.

Explore away, and be prepared to tweak it to get what you want.

The julias can be interesting with the right radian measures too.

Just realised that this form is the same as that derived for the helical structures used in the thread spacetime manifold candidates (http://www.fractalforums.com/complex-numbers/the-space-time-manifold-tensor-candidates/new/#new).

This makes me think of Fractalwoman (http://www.fractalforums.com/meet-and-greet/fractalcosmology/) and a comment she made about using the mandelbrot for cosmology.

(http://www.forkosh.dreamhost.com/mimetex.cgi?z%20-%3E%20z^2%20+%20c%20=%20e^{arg(z)%20%5Ccdot%202%20%5Ccdot%20i}%20%5Ccdot%20abs(z)^2%20+%20c)

(http://www.forkosh.dreamhost.com/mimetex.cgi?z%20-%3E%20e^{(arg(z)+n)%20%5Ccdot%20i}%20%5Ccdot%20(abs(Z)%20+%20n)%20+%20c)

Two equations found here (http://www.fractalforums.com/new-theories-and-research/yet-another-alternate-transformation-idea/msg18046/#new)

So here are two pics based on the first form, one with a quad r^2 and the other with a real r^2

(http://www.fractalforums.com/gallery/2/410_09_06_10_11_13_39_1.png)  (http://www.fractalforums.com/gallery/2/410_09_06_10_11_13_39_0.png)

Here are some pics.
This julia was interesting cos of the bumps
(http://www.fractalforums.com/gallery/2/410_09_06_10_11_13_39_2.png)
This julia was a variation
(http://www.fractalforums.com/gallery/2/410_09_06_10_11_13_39_3.png)
This julia uses sin for the j coefficent. I have not got round to using j for the exp imaginary term in relation to this.
(http://www.fractalforums.com/gallery/2/410_09_06_10_11_13_39_4.png)

All these were produced using Terry gintz Quasz for mac, but i used a quad form of r^2 by mistake as |z| in quasz is a quad form.

I will have to wait for my new computer to use the windows version, but the mac version is so cool!

Define exponentialish functions by

expish(x,y,z) = (exp(x)*(cos(y)*f1(x,y,z), sin(y)*f2(x,y,z), f3(x,y,z)),
where f1(x,y,0) = f2(x,y,0) = 1 and f3(x,y,0) = 0.

Similarly, define logarithmish functions by
logish(x,y,z) = (ln(x^2+y^2) + g1(x,y,z), atan2(x+iy)+g2(x,y,z), g3(x,y,z)),
where g1(x,y,0) = g2(x,y,0) = g3(x,y,0) = 0.

Here is what you get for
     
      expish(2*logish(z))+c

for
f1(x,y,z) = f2(x,y,z) = cos(z), f3=sin(z):
g1 = ln(x^2+y^2+z^2) - ln(x^2+y^2), g2=0, g3 = asin(z/r).


This form is found and developed here by schlega (http://www.fractalforums.com/3d-fractal-generation/exponentialish-extensions-of-the-mandelbrot-set/)