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RuneLancer

Attractors - Display methods

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Seeing as a few questions pertaining to fractals have crept up in here, I''m guessing it''s the right place to ask about this. I''ve always had an intense interest in fractals, attractors, l-systems, IFSes and other chaotic systems that produce imagery. Lately, attractors are what have held my interest the most. I tried something out a little while back on a whim and got some really nice results. Basically, I attempted different plotting methods instead of the typical "color the dot where the X and Y parameters point to" method. I don''t have access to my programs here since I''m at college, in an SQL class, being bored. But it went something a little like using the on-screen position as the starting X and Y values and coloring the pixels according to various things such as how far they ended up from their initial location after n iterations or how many iterations it took before they tended towards infinity or enthropy. A little like the more standard fractals (mandelbroth, julia...). I''ll try to get some screenshots up when I get home but what I''m curious about is... anyone else has tried changing known chaotic systems and gotten interesting results? Just out of curiousity and for fun (as I can''t quite think of any application of what I''ve produced other than eye-candy and amusement ).

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In a similar vein.. I made a program to estimate the Box Counting Hausdorff dimension of an image. It would generate the boundary of the plotted fractal set, and output the dimension. I then plotted the fractal dimension as grey intensity (scaled appropriately of course) against the Julia set seed value... and the result was VERY interesting (though it took a LONG time to plot. A 320x240 plot took about 25hours to plot on my 1.4Ghz laptop). The result was a greyscale image, whose outline was (as is to be expected) the Mandelbrot set... but within this region, there appeared to be many copies of the major features visible in the Julia set near the seed location. I have never seen any reference to this being done before, and could come up with no explanation for it (My lecturers at the time found it very interesting and could not even hypothesise a reason for the structures present).

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25 hours on a 1.4Ghz o.o Good god, I hope you saved a screenie of that! (And if you have, mind posting it? I''m really curious as to what this looks like...) I don''t even want to ponder how long it would take on my PIII 550...

Though it is well-known that Mandelbroth and Julia are linked (well, technically, the same but drawn using a different method; Julia uses a fixed "seed" value whereas Mandelbroth uses the position of the current pixel as the "seed" value), I can''t think of a reason why this would happen. Maybe I''m just thinking along the wrong lines though... There might be a relation between the Hausdorff dimension and the initial "seed" of the fractal since the Julia fractal varies strongly in terms of occupied space with different parameters, though to be honest the Hausdorff dimension is something I only partly understand... My guess is that the reason it behaves that way is because you''re doing something a little similare to what Mandelbroth did (a ''map'' of every Julia set by using the position in a complex plane as the "seed" value for the fractal).

As promised, here''s a screenshot of what I was talking about.



Unfortunatly, I have no clue what the equation I used for the attractor that generated this image was. It definitively wasn''t something well known (Hénon, Lorenz...), that much I remember. It was a home-brewed formula that gave nice results, though. I could code the program again and try feeding it Lorenz maybe, that always gives interesting results...

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I did it as part of my dissertation, so it must be about somewhere (I guess)... not here though, since I''m not on my laptop. I''ll have a look around for it when I am reunitied with my laptop, and then I''ll post a pic. The reason its outline is a mandelbrot set is clear, from the definition of the mandelbrot set, as the set of all connected Julia sets. The interesting part is the content of the set. Hopefully I will find it, and you can see what I mean .

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