Coolness! Even though it has whipped cream / taffy, I like this idea. Thanks for the lesson.
In general, I think the taffy problem happens because the power function isn't conformal. The 3D White-Nylander power functions are least conformal near the poles. In experimenting with 2D escape time fractals over the years, I think I've noticed that fractals that look all stretched out come from formulas that aren't conformal, and fractals that look "nice" invariably come from formulas that are. I think this makes sense because that's what a non-conformal map does--it "stretches out" space.
As pointed out by Tglad on another thread, Liouville's theorem says that the only smooth conformal maps possible in > 2 dimensions are Moebius transformations, which unfortunately don't become complicated when iterated since they form a closed group. I suspect that if one could find a 3D map that was somehow "close" to being conformal, it still might not be good enough because any deviations from conformality might get magnified when iterated lots of times. I think completely getting rid of the taffy is likely a hopeless task, but there may be other nice ways to compromise. In my opinion, the high-degree Mandelbulb fractals are nice ones because they "spread the taffy around" so that there are lots of areas that aren't dominated by it. It seems kinda hard to find much variety in them though from what I've seen so far.
Just out of curiosity, and because you KNOW I'm going to have to go now and find a 3d conformal function, can you explain in what sense the 2d mandelbrot function is conformal? I read on wikipedia that conformal functions preserve angles; if I may ask, which angle is being preserved by the 2d mandelbrot function?
January 02, 2010, 06:31:13 PM
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