Area of surface of bulb is the product of the rotation of a 2d Mandelbrot around a 2d Mandelbrot (both with Hausdorff of 2).
The mandelbulb is not the product of 2 Mandelbrot sets.
Here is some code (should make it clear, since this shows how the bulb works ... like the regular 2d set):
sx,sy,sz are starting x,y, and z components; nx, ny, and nz are calculated components before pixel component addition;
r1=sqrt(sqr(sy)+sqr(sz));
r3=r1^-n; //take it to the negative power to avoid division by zero error!
victor=complex(sx,r1); // complex (x,y) is ChaosPro's command to create a complex number out of 2 real components x and y
bravo=complex(sy,sz); // the second part (1st,2nd) is the imaginary component
victor=victor^n;
bravo=bravo^n;
nx=real (victor);
ny=imag (victor) * real (bravo) *r3; // you have to divide out the magnitude (r3=r1^n) because it has already been applied
nz=imag (victor) * imag (bravo) *r3; // twice, once computing victor, once computing bravo
then add in your pixel values..
So you can see we have something that resembles two 2d sets influencing one another, almost like it's a product of 2 2d sets....
Wouldn't this make the Hausdorff 4?
Certainly not, you can't just add dimensions. And, it can't be more than the space in which it lies.
So it's maximum hausdorff is 3, and it has to be 3.. although it has a higher apparent complexity than a spun brot...
March 26, 2007, 05:48:56 AM
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