Creative 3d mandelbrot formulae

[jehovajah]

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.

[jehovajah]

Just realised that this form is the same as that derived for the helical structures used in the thread spacetime manifold candidates.

This makes me think of Fractalwoman and a comment she made about using the mandelbrot for cosmology.

[jehovajah]

Two equations found here

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

 

[jehovajah]

Here are some pics.
This julia was interesting cos of the bumps

This julia was a variation

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.


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!

[jehovajah]

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