![]() ![]() ![]() Looking at Devaney's images, I'd guess that the reason why these fractals have the beautiful Sierpinski-gasket-like structures, while the standard quadratic Julia and Mandelbrot sets don't, is that each of the formulas defining these fractals is a rational function, that is, a ratio of two polynomials, rather than a single polynomial. I'm by no means knowledgeable on this subject, but I've been looking at some of Robert Devaney's papers, which I came across via. The formulas for the two fractals are also given there. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals. In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. Ultra fractal apollonian gasket software#It quotes the Fractint documentation (I can't resist mentioning that Fractint is the grand-daddy of freeware fractal-generating software for personal computers - it had its first release in 1988, and is still being maintained!): I searched Google for "magnetic fractal", and found the answer on the first hit. ![]()
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