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All,

 Thanks for the compliments. This is an interesting and encouraging discovery, but the crown jewel has yet to be discovered.
 
 Twinbee: I don't think zooming in this Julia set will bring anything exciting. If you consider the 2-D Julia variant, the zooming in just brings in more and more smaller bulbs, with no mixing of antennae-like structures.

 One interesting application of this "mirroring" method is that it produces more aesthetically pleasing 3-D Julia set renderings by removing the z=+/-1 poles. The mirroring is continuous, but not analytic.

 I experimented with some variants to fix the squashing of the bulbs that start in the x=0 plane. Here's the latest formula:

 if (z*z<y*y) {                                                                                                                     
  r = x*x+y*y+z*z+4*y*y*z*z;
  r1 = sqrt(r);
  theta = 2*atan2(y,x);
  phi = 2*asin(z/r1);
  r = x*x+y*y+z*z;
  x1 = r*cos(theta)*cos(phi);
  y1 = r*sin(theta)*cos(phi);
  z1 = -r*sin(phi);                                                                                                   
 } else {                                                                     
  r = x*x+y*y+z*z+4*y*y*z*z;
  r1 = sqrt(r);
  theta = 2*atan2(z,x);
  phi = 2*asin(y/r1);
  r = x*x+y*y+z*z;
  x1 = r*cos(theta)*cos(phi);
  y1 = -r*sin(phi);
  z1 = r*sin(theta)*cos(phi);                                                                                                                       
 }
x=x1+a;y=y1+b;z=z1+c;

The Julia variant (-0.8,0,0) looks a little less distorted than the previous one:


But the M-set lacks the beauty and symmetry of the J-set:

I'd be really interested in seeing a "julibrot" projection of that last one....

It's surreal!, but almost everything here is

I like it a lot anyway...

Spectacular! Wow, how close is that? Distortion on the spheres is either fixed, or very minimal for the smaller spheres maybe.

EDIT: Wouldn't mind seeing it with a lower iteration...

Various maximum iterations. My numbering scheme may be off by 1:



The code:

  psi = atan2(z,y)+pi*2;
  psi2 = 0;
  while (psi > pi/8) {
   psi -= pi/4;
   psi2 -= pi/4;
  }
  cs = cos(psi2);
  sn = sin(psi2);
  y1 = y*cs-z*sn;
  z1 = y*sn+z*cs;
  y = y1; z = z1;

  x1 = (x*x-y*y)*(1-z*z/(x*x+y*y+z*z));
  y1 =   (2*x*y)*(1-z*z/(x*x+y*y+z*z));
  z1 = 2*z*sqrt(x*x+y*y);

  y = y1*cs+z1*sn;
  z = -y1*sn+z1*cs;

   x=x1-0.8;
   y=y+0;
   z=z+0;

Note the slight change in the Mandelbulb equation which seems to cause less distortion, also z1 can be positive or negative yet produces the same Julia set.

Because of my success with the Julia variant, I've been obsessed on cracking the code using symmetry operations to cull out the right stuff.  I've found three M-set  variants so far.

The following is the closest hit, but still a far way off. The general idea is to vary between + and - rsin(phi) over even and odd iterations.

The 16-fold symmetry was done for a specific reason. The un-symmetrized version has a little bulb in a small angular span. The symmetry operation was used to extract that out but discard the boring stuff.

I conjecture that the real M3D, if it exists, requires minimal or no symmetrizing.



I'm using a very high number of iterations (N=200). If you dial that down, you lose the smooth bulbs. But, maybe, there would be less fuzz and more of something to discern when zooming in. The bulbs look promising from a distance, but upon zooming in lack the symmetry seen in the Julia set.

It keeps getting closer! Good for you on not giving up - the Mandelbulb would look like a barren desert compared to the delights we can expect if you found it.

I agree that one wouldn't think any symmetry operations to be needed. Actually, I wonder if you could zoom into these areas highlighted by red boxes - we could already start to be seeing some strange and wonderful things.

EDIT: Just read your end quote. To be honest, I'd still like to see some zoom-ins to the earlier Julia version. I know the Julia has less interesting detail than the Mandelbrot equivalent generally, but I'm still curious