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Samith

Julia Sets - how do they work if the constant doesn't change?

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Ok, Julia sets are really starting to bother me. From what I know, there is the Mandelbrot Set which is just znew = zold^2 + c, where c is the location of the pixel on the complex plane, and you reiterate the series until |znew| > 2, and you do this for each pixel. But, the Julia set, as far as I know, is just znew = zold^2 + c, but c is a fixed constant and it never changes! How in the world do you get a fractal from this? To me it seems like you would just get.....well....just one pixel from the Mandelbrot Set. Well, obviously I''m wrong, so, does anyone want to clarify?

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The julia set is the non-divergent points under the iteration you outline. The reason All points do not give the same value, is that the initial value (your z0) is determined by the coordinate point you wish to check. This demonstrates that slightly perturbing hte initial value (z0 shifts slightly) can completely alter the long term behaviour of the resultant point (convergent, or divergent). This is the essence of chaos (or more specifically, sensitive dependance on initial conditions). The Mandelbrot set is a pure mathematical set, is the set of all "connected" (this has very specific mathematical/analytic meaning btw) Julia sets. Hope this helps .

EDIT: Just to point out also, the reason you check if they go over 2 in magnitude... is because for any given seed... if the modulus goes over 2.. the value will necessarily diverge.

[edited by - dmounty on October 21, 2003 6:50:11 PM]

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Ok, so are these Julia sets like cross sections of a 4d Mandelbrot Set? I can''t really explain how I came to this conclusion, but it makes sense....somewhat....in my mind!

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quote:
Original post by Samith
Ok, so are these Julia sets like cross sections of a 4d Mandelbrot Set? I can''t really explain how I came to this conclusion, but it makes sense....somewhat....in my mind!


In a sense, it''s sorta that.

The Mandelbroth set is an attempt to display every possible Julia set by using a complex plane as a parameter instead of a single position. That is, instead of using say... 0.62334-0.22135i, you use the whole range of values between -2-2i and 2+2i. (Or 1? Ah, hrm, a little rusty, hehe...). For this reason, any point in the Mandelbroth fractal that looks "interesting" (ie, not the actual set (the part typically colored black) or a "bassin", but tangles of colors) will generate an interesting Julia fractal if used as its parameters.

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