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Author Topic: exponential-based complex dynamics  (Read 2527 times)
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bh
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« on: September 15, 2007, 08:08:37 PM »

Three images I submitted to the Fractal Art Contest 2007, but which didn't get much success there:


http://i16.tinypic.com/4uet7o2.jpg


http://i15.tinypic.com/6chc5ti.jpg


http://i16.tinypic.com/2ilfrr9.jpg

I am happy with the shapes I obtained. They have patterns characteristic of the dynamics of complex functions with exponentials. Such images are quite rare on the Web; I found only a few examples, like I think this image by Titia (very good work, by the way).

On a side note, I want to say that the contest organizers did a bad job compressing my fractals from the 1600x1600 images I provided. They applied the standard but sometimes terrible 2x2 chroma subsampling. Just compare:

http://www.fractalartcontests.com/2007/showentry.php?entryid=119
http://i4.tinypic.com/6d1e8lt.jpg
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bh
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« Reply #1 on: September 16, 2007, 05:06:29 PM »

very nice renders! are you using a function like z <- a * e^(bi + c) + d, or something similar?

It was:

f(z) =  z * exp((1-z^p)/(p*(1+c*z^p)))

with p = 3, and

c = 0.0539 + 0.1629*I
c = 0.0047 + 0.2087*I
c = 0.0139167 + 0.111517*I
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gandreas
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« Reply #2 on: September 17, 2007, 04:32:08 PM »

Especially nice considering how often the "fingers" get in the way of fractals like that (and if you increase the bailout to try to reduce that effect, you start getting moire effects, and eventually end up with Cantor dust since it's "fingers all the way down").

For a good introduction to the "BRD" set (and some other pretty pictures), check out:

"Growth in Complex Exponential Dynamics"
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bh
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« Reply #3 on: September 19, 2007, 11:23:56 PM »

thanks, i may have to try that one sometime soon smiley does it have a name?

It might be that I am the inventor of this particular formula smiley
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bh
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« Reply #4 on: September 19, 2007, 11:52:26 PM »

Especially nice considering how often the "fingers" get in the way of fractals like that (and if you increase the bailout to try to reduce that effect, you start getting moire effects, and eventually end up with Cantor dust since it's "fingers all the way down").

For a good introduction to the "BRD" set (and some other pretty pictures), check out:

"Growth in Complex Exponential Dynamics"


Hi Gandreas, thanks a lot for the link.

Actually, my formula is much friendlier than those in the article, like f(z) = lambda * exp(z). In my formula, if z is large, the formula is about like f(z) = z * exp(-1/(p*c)), which, assuming Re(c) > 0, makes z smaller; and if z is small, the formula is about like f(z) = z * exp(1/p), which makes z larger.

I said that my images are characteristic of the dynamics of functions with complex exponentials, but perhaps I should restrain my words. One thing I found characteristic of a certain class of formulas are the points where we see an infinite number of branches originating, and with a clear-cut separation along a line. These singularities appear because of the poles of the function w = (1-z^p)/(p*(1+c*z^p)); near the pole, w will be very large in absolute value, either with real part positive or negative, and thus exp(w) will be either very small or very large.

My images were still a challenge regarding anti-aliasing. I found that sometimes I needed 16x16 supersampling to get adequate results.
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lkmitch
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« Reply #5 on: September 20, 2007, 07:35:40 PM »

Such images are quite rare on the Web

Here's one of mine:

http://www.fractalus.com/kerry/gallery13/ladder.html

I think there are a couple of reasons why we don't see more exponential fractals:

  • The libraries for evaluating complex exponentials (including trig functions) are much slower than arithmetic computations
  • The standard bailout condition based on |z| doesn't usually work, particularly with z = c * exp(z).  Depending on the function, it can take some fiddling to determine the best bailout condition.

Both of these mean that exponential images can take orders of magnitude longer to compute than fractals based on polynomial functions. So maybe the tradeoff in taking more time is that using exponentials is one way to distinguish yourself as a fractal artist.

Kerry
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David Makin
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« Reply #6 on: September 21, 2007, 09:27:21 AM »

I think one of the reasons exp formulas are used less often is simply that there are fewer colourings that work well with them and you are more reliant on the shape of the actual fractal - a lot of folks seem happier playing with the colourings than with the main formula smiley
Same goes for the trig functions.
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lkmitch
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« Reply #7 on: September 21, 2007, 06:41:29 PM »

  • The standard bailout condition based on |z| doesn't usually work, particularly with z = c * exp(z).  Depending on the function, it can take some fiddling to determine the best bailout condition.

i suppose one could check ln|z|, or equivalently check against a magnitude like exp(usual_bailout).

Checking ln|z| would help with overflow, but the basic issue is that you can have spires and structures that extend to infinity, meaning that points in the same neighborhood can have orbits with arbitrarily large magnitudes and either diverge or stay in the set.  That's why a lot of colorings don't work for exponentials--most colorings are designed assuming that the orbit points are somewhat even distributed around the complex plane, or assuming that a large pixel value will lead to an orbit with large values.  And since, in the complex world, trigonometric functions are exponential functions, trig functions don't often work well, either.

Kerry
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bh
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« Reply #8 on: September 21, 2007, 09:15:41 PM »

I think one of the reasons exp formulas are used less often is simply that there are fewer colourings that work well with them and you are more reliant on the shape of the actual fractal - a lot of folks seem happier playing with the colourings than with the main formula smiley
Same goes for the trig functions.

Quite true.
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gandreas
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« Reply #9 on: September 27, 2007, 12:04:37 AM »

for example, the recent apophysis 3d hack leverages the huge variety of fractals associated with flame rendering[2] and is reasonably flexible wrt colouring, however what you have in the end is nebulous clouds (unless you're really careful with your viewing angle etc). here's an example rendered with chaoscope: http://silent--dreamer.deviantart.com/art/Plasmic-Tornado-2-15520497

When I added 3D flames into quadrium | flame, the first thing I noticed was how cool things like a "julian badge" looked when extended to 3D as you twisted it a round or tweaked the parameters - right up until the point that you stopped manipulation the view space, at which point it turned flat.  If I made a movie of one spinning & changing, though, it was great.  Basically, the human visual system is really good at deducing 3D information from moving abstract points, but once those points stop moving, it falls flat.  For a similar effect, take a look at the movie at http://www.lifesci.sussex.ac.uk/home/George_Mather/Motion/BM.HTML - it's quite clear what you're seeing as long as it is moving, pause it and you just see a handful of dots.

Without some sort of depth cueing, non-animated 3D flame fractals are sadly somewhat less interesting than they really seem like they should be...
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