Ever wonder what a real, three-dimensional Mandelbrot-type fractal might look like?
Well, here’s a site devoted to one possibility: Mandelbulb.
Appearance-wise, it doesn’t look all that much like a 3D Mandelbrot (except when you look closely in some spots—see a subheading on the 2nd page of the site for more about this issue), but in terms of complexity, it’s apparently the best 3D approach to the infinite complexity of repetition and variation seen in the Mandelbrot set that anyone has yet found. The function used is very similar to the Mandelbrot set function, except z is raised to the 8th power on each iteration, rather than simply squared. Also, due to the fact that three coordinates are needed on each point, rather than just two, the exponentiation of the coordinates is not the same as the simple complex-number multiplication used for the Mandelbrot set (this is really the central issue with finding a “3D Mandelbrot set”—the problem is finding an analog to complex multiplication that can be used to define points in space, rather than just on a plane). Details on the formula can be found here.
However, the main reason I am linking this is because of the pictures. There are some really fascinating, beautiful pictures. Wow. Definitely a must see—I commented on Flickr this morning that it’s the most interesting thing I’ve seen on the internet in quite a long time. I’m tempted to hotlink one of the smaller images…but that would be rude. Besides, the one I pick might not be the one you like best anyway. So just go there already. :)