Another 3-D mandelbrot attempt

[msltoe]

Figure out what I'm trying to do here. I'll explain later...

 First part:
   r = x*x+y*y+z*z;
   r1 = sqrt(r);
   phi = asin(z/r1);
   r2 = 1.0/sqrt(fabs(cos(phi)));

   if (r2>10){x=1000;} else { // Don't bother at phi near pi/2.

   phi = r2 * phi;
   r1 = pow(r,0.5*r2*r2);
   
   r3 = sqrt(x*x+y*y);
   x1 = r1*x/r3*cos(phi);
   y1 = r1*y/r3*cos(phi);
   z1 = r1*sin(phi);
   x=x1;y=y1;z=z1;

 Second part:
   x1 = (x*x-y*y);
   y1 = (2*x*y);
   z1 = 2*z*sqrt(x*x+y*y+z*z);
   
   x = x1+a; y = y1+b; z = z1+c;
  }

Also, I transform a,b,c at the start using a similar transform as the first part.

Here's the M-set:

[msltoe]

A J-set at (0, -0.65, 0):

[Alef]

It reminded me of chameleon eyes:

A Mandelchameleon;) Could you make larger powers. Power 3 shape in 3D somewhat reminds two headed turtle, maybe power 3 simmetry fitts better a 3D.

[kram1032]

very interesting. Instead of whipped cream, we get what looks like "perfect" smoothness.
Certainly nice.

[msltoe]

kram1032: I like your analysis.
Alef: pretty pic. What's the formula?

What I think is missing is a little more density of the bulb parts.
I've tried a few tricks like reflections in phi space, but nothing appears as clean as the original.

[jehovajah]

Very nice and what i felt you would be able to do!

A little texture  in the colouring algorithm maybe, but who can get any closer to the spherical trochoids? May be Matt might have an idea or two!