Title: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: Timeroot on January 04, 2010, 06:11:33 AM 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? Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: David Makin on January 04, 2010, 12:48:35 PM 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). Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: Timeroot on January 05, 2010, 12:58:20 AM Okay, thanks! That sounds like pretty much it, I just didn't know the name for it or anything. ;D
Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: David Makin on January 05, 2010, 01:16:51 AM Just thought you may like to see this.....3D slices of the 4D Julibrot for z^2+c.
http://www.youtube.com/watch?v=gr-ul7sZDwc (http://www.youtube.com/watch?v=gr-ul7sZDwc) Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: Melancholyman on January 05, 2010, 02:18:13 AM Looks like a porcupine I think female brots, are rather evil! :dink:
Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: kram1032 on January 05, 2010, 02:58:53 AM Well, Julia refers to a surname rather than a first name :P
Title: Re: Skewed Slices of 4D Mandelbrot/Julia fractal Post by: Nahee_Enterprises on January 05, 2010, 08:41:34 AM 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. ..... Unfortunately, I can't get anything to work in FractInt. Are you familiar with the various formulae of Jim Muth?? He tends to create special ones for viewing from various angles, and categorizes them with names like:
In fact, somewhere in his numerous FOTD postings he explains them in detail, along with several examples of images: http://www.Nahee.com/FOTD/ (http://www.Nahee.com/FOTD/) |