Okay, so you know how the Mandelbrot set and the Julia set are just two slices of the same 4D fractal? This is how it works: Each pixel of the Mandelbrot set is identical to the center pixel of the Julia set with that parameter. Similarly, each pixel of the (0,0) Julia set is identical to the center pixel of the Mandelbrot set with that offset - Each pixel of the Mandelbrot set with an offset of (.2,.1) is identical to the (.2,.1) pixel of the Julia set with that parameter. Basically, they're just two different slices of the same 4D fractal. Each co-ordinate (k,l,m,n) of this fractal is iterated (using the Fractint syntax):
z=k+i*l, c=m+i*n:
z=z*z+c
|z|=4
Then, in the Mandelbrot set, m=x, and n=y (where x and y are the co-ordinates of the screen pixel), while k and l are set to 0 by default, and can be changed by introducing an offset. For the Julia set, on the other hand, m and n are set by the parameter, while k=x and l=y.
What I'm wondering is how it would be possible to view this 4D fractal with different slices. For example, imagine we were to set m and k to a parameter, while letting n and l be dependent upon the position of the screen. Or even, we could take the slice at an angle - imagine k=0.5x+2.1, l=0.9y-0.1, m=2x-0.2y+1, and n=0.9x+0.1y-0.5. This may need some slight modifications to give it proper scaling and remove any skew, but you get the idea (I hope). Unfortunately, I can't get anything to work in Fractint. Any help/comments/criticism here?
If I understand correctly you're just suggesting rendering any 2D slice from the 4D Julibrot, while it doesn't allow every possiblitiy my formula for Ultra Fractal ->mmf.ufm->4D Transform lets you do that for a slice at any 3D angle, though the slices are restricted to flat planes.
The formula is also compatible with ChaosPro (freeware).
February 28, 2012, 05:09:50 PM
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