3d Mandelbrot attempt

[Jesse]

  UPDATE    Corrected the formulas.  Sorry Jesse, realized we should add an x-pixel component part to the sign check- it really improves the quality of the fractal on the -x axis side, in my opinion.  Might need to look a bit closer.. but ehh....

Yes, now it is complete!  Very good one!!!

The formula for m3d to download is here:
http://www.fractalforums.com/index.php?topic=6061.0

 

[M Benesi]

  Nice implementation!

  Lol.. you were looking close to where I was lookin' last night:

  Click to biggify:

[M Benesi]

  For all n z^n other than 2,6,10.... you can use a similar formula to the previous trig formula I posted in order to get a slightly more dynamic fractal.

  Just eliminate the x vs. r1 check to keep the fractal continuous.  Makes the other fractals... have nice loopy areas.

Code:

r=(x^2+y^2+z^2) ^ (n/2);
r2=sqrt (y^2+z^2);

theta= atan2 [x + i r2];   //  atan2 is equivalent of arg... I think... or something like that.... 
phi=    atan2  [y+ i z];

newx= r * cos [theta*n];

// REMOVE THIS PART:   if (x>r2  && pixelr>0) {newx=-newx;}              

r=-r* |sin [theta*n] |;   // still keep the absolute value here   
                               
newy= r * cos [phi*n];
newz= r * sin [phi*n];     //changing either the z or y sign results in a slightly different fractal, if I recall correctly it has a large cream section

.... so now add in yur pixel components, or maybe yer julia components, or do whatever.  Enjoy.  

  Basically, constraining the one components to a specific sign allows the y vs. z Mandelbrot code portion to keep its sign variation, instead of being influenced by the other portions sign as well (which results in chaos at lower n, although by z^8... it's not as big of an issue).

[Tater]

 So basically, I dug out an old idea, and apparently found a realllllllly good way to apply it.  

  The thing was, for a long time, I had this gut feeling that it was the sign assignment that was holding us away from fractally goodness for z^2.  I attempted many different methods, but not this one (and this one can be applied to the other formulas as well, one in particular comes to mind, the one with the perfect 2d cross section).  Anyways...

  I did my standard trick- double complex triplex (use complex numbers instead of trig- it runs about 2 times as fast on my old comp), using my favorite formula:  x vs. [y vs. z]   (the other formula that comes to mind is the [x vs. y] vs. z formula).  

  This time, I constrained the x variable to absolute value, but allowed sign variation on the y and z.  So basically, the new x value (before adding in pixel component) is |new x value|.

  Anyways, here are a few images.  As you will see, it has a few different types of forms... well, a lot.  There is variety... weird shapes and the like.  There are seahorse spins.  There are... other things we've caught glimpses of in other attempts to hit the true deal....  Anyways, nothing that great yet, but I wanted to put it out there.  Haven't even tried the other z^n yet.

Another way you can get that kind of detail from power n = 2 on the radius of the vector is  by making the angles change at a different, but related, rates. For instance, let n = 2 be the power applied to the length of the vector, and multiply phi by 1.5*n = 3 and theta by -1.5 n =-3.

[bib]

I had missed this thread. Reminds me the Thornton variant:


<a href="http://www.youtube.com/v/m0Yx3h8rvl0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/m0Yx3h8rvl0&rel=1&fs=1&hd=1</a>

[M Benesi]

Another way you can get that kind of detail from power n = 2 on the radius of the vector is  by making the angles change at a different, but related, rates. For instance, let n = 2 be the power applied to the length of the vector, and multiply phi by 1.5*n = 3 and theta by -1.5 n =-3.

  Well, that can be done, but it isn't a strict power 2 in some sense.  The odd powers (3,5,7,9....) don't have the sign variance that the even powers do, which is why using the 3rd power for the phi (y/z) portion doesn't generate crappy chaos. 


  @ bib-  Yeah it does <clicky clicky>, but that formula variant (which this may be based upon) didn't produce nice Mandelbrots - only Julias, ehh?