Another Nonworking Attempt

[Geometrian]

Hi,

For a 2D Mandelbrot Set expressed in 3D (where X, Y, and Z correspond to the complex plane and height, respectively) the axis of rotation (of the complex numbers expressed in polar form) is [0.0,0.0,1.0].  So, to square Z_n, you simply rotate it around that 3D axis.  This produces a "stack" of Mandelbrot sets along the z axis. 

I had the idea of taking the cross product of the major axis (X) and a vector to the current point to get a new axis of rotation.  Notice that, plotting in X, Y, Z, for Z=0, the axis is [0.0,0.0,1.0]; thus, the resultant solid will have a cross section at Z=0 of the Mandelbrot set.  However, as Z changes, the axis of rotation changes. 

After resuscitating some of my old GPU raytracing code, (which I have since rewritten, but didn't want to merge), I was able to plot the thing.  First the relevant part of the (GLSL) source, which should clarify the above explanation:

Code:

vec3 coord0 = vec3(x,y,z);
vec3 coordn = coord0;
mat3 rot_matrix;
float l;
for (int i=0;i<fractal_iter;i++) {
    rot_matrix = rotation_matrix_arbitrary( abs(normalize(cross(coordn,vec3(1.0,0.0,0.0)))), -atan2(coordn.y,coordn.x) );
    l = length(coordn);
    coordn = rot_matrix * coordn;
    coordn = l*coordn;
    coordn += coord0;
    if (length(coordn)>2.0) { return 0.0; }
}

"rotation_matrix_arbitrary" finds a matrix that rotates around an axis specified by the first argument by an angle specified by the second argument.  "atan2" must be hardcoded, because it doesn't exist natively in GLSL.

The result, unfortunately, is not a satisfactory 3D Mandelbrot Set, but I present it here for the interested.
Again, notice that the cross section of the thing is the Mandelbrot Set, but that it's not really a 3D fractal.

Ian

[kram1032]

It' certainly a nice shape

[KRAFTWERK]

Yes, pretty cool!