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Lyapunov 01 by BernardH Lyapunov 01 by BernardH
Once again, an old image, but I just discovered that it got featured and made the front page of Wikipedia on September 13: [link], so I'm going to post it here. A few words of explanation:

1) I have only done the rendering of this image. The original designer was a user called Wickerprints on Wikipedia.

2) The original rendering by Wickerprints was, with regard to antialiasing, well, horrible. The rendering I made at the time was a bit better. The rendering here is one I made 2 years ago and I tried to make the quality as good as I could. I will have to upload it in place of the featured image on Wikipedia.

3) Lyapunov fractals take longer to render than the classical Julia or Mandelbrot fractals, because many points have a chaotic behaviour and so the loop cannot be exited early.

4) Lyapunov fractals also present a big challenge concerning antialiasing. You should understand why if you look at the image in full size. The image here has, if I remember right, 32x32 multisampling antialiasing. Actually, I'm not perfectly happy with the antialiasing; it has a flaw: the narrow strokes of less than a pixel wide appear crenellated. This is one case where the antialiasing should not be the naive, regular multisampling, but should use some kind of filter... I don't know exactly yet.

5) Lyapunov fractals are much like Mandelbrot fractals, except that, instead of one complex parameters, they have two real ones. For any of the bands in the image, you can think of its cross section as the real part of the standard Mandelbrot set. A point at the start of the section can be mapped to c=0.25; the point at the intersection with the first dark line is c=0; the next bright point is c=-0.75, the next dark point is c=-1, the next bright point is c=-1.25...

6) The iteration function can easily be seen to have 12 critical points (points where the derivative is zero). Therefore, by a classical argument, the dynamics for any point in the parameter space could have up to 12 attractive cycles. However, when we render a Lyapunov fractal in the usual way, we are only iterating from one of those critical points... and this is a flaw, we really should iterate from all of them (which means 12 times as much work) to identify all the attractive cycles, and then find another way to determine which one will drive the coloring; a way that depends on the intrinsic properties of the attractive cycle and not on how we got there. What are the practical consequences of this flaw? Well, if you look at the image in detail, you should find many places where a yellow piece disappears in mid air with no apparent reason, or gets covered by another one in a way that seems inconsistent and sometimes even chaotic. It is not a nice feature and it should be avoided.
kram1032 Featured By Owner Sep 15, 2012
Haha, nice. Just by chance, I've seen it, like, half a week ago on there.
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Submitted on
September 14, 2012
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