Just a question - i see a lot of stuff on the net, that juxtapose fractals next to chaos theory, etc. are fractals really chaotic? they seem to be the ultimate expression of order. they are created from mathematical equations & are therefore inherently predictable. what is all this chaos charade about anyway? i feel that there is no such thing as chaos - it is all rational & ultimately ordered. as programmers are aware there is no such thing as a truly random number. chaos is just a blanket form to describe the ineffable. only upon looking closer does the chaos coalesce into armies of order. numbers crunching off into the horizon. speaking of which, my bed beckons :oP

Fractals are both chaotic and ordered, as chaos (in the sense of fractals) is order. Basically, small changes in the input lead to big changes in the output, in ways that may seem random but are fundamentally well-ordered. In the typical fractal (like the Mandelbrot set), a small change in c may take you from a fixed point orbit to a divergent orbit or from a periodic orbit to a chaotic orbit. The fact that there's infinite detail in a fractal suggests that it is chaotic, and the fact that one can see repeating structures in that detail suggests that its ordered. Kerry

An interesting point. just to clarify - are you using the word "chaos" in a mathematical sense? "Chaos = A dynamical system that has a sensitive dependence on its initial conditions." regarding the last part of your response, is it fair to say that if something has infinite detail, that it is chaotic? Surely, given infinite time & the desire, the order can be ultimately worked out? regardless (of this longwinded approach), the fact that you know the formula indicates that detail follows a fixed rule & therefore can be nothing but ordered. I must admit to not being aware of the detailed mathematics behind fractals, so please set me straight if i'm confusing the issue. Thanks GDV

To be precise, chaotic is a classification of a dynamical system (like iterating z = z^2 + c), and a fractal is a shape. Quite often, fractals belie chaos in the underlying dynamical system, but there are chaotic systems that don't give rise to fractals, and fractals that don't represent chaos. But usually, the two go together. The fact that the fractal comes from a formula means that it is deterministic, which is a requirement of chaos. However, chaotic systems have order to them that is not apparent from the formulas. Usually, that order becomes clear when plotting the dynamics of the system in the "right" way, and it can also be revealed through statistical analyses of the data. Kerry

thanks v much for the info kerry - think you & another friend pointed me in the right direction. some other links that were helpful in understanding what you meant were: http://order.ph.utexas.edu/chaos/index.html & http://www.duke.edu/~mjd/chaos/chaosh.html cheers! regards GDV

I ordered my fractals off the Internet but they never came, so I had to make my own. I always thought the chaos theory was about making an order out of chaos, therefore it was a little of both. Fractals seem to fit right in with it. Now I'm completely confused.

The way I look at it: The pictures that we create with all of our various programs, are merely ways to visualize what is in fact a huge collection of numbers, generated by the equations. The various coloring algorithms merely let us 'see' different relationships between groupings of those numbers. Thus, when using your basic Mandelbrot set, you can have an infinite numbers of pictures resulting, dependant on whether you use a 'distance' coloring algorithm or a 'real numbers' coloring, etc.
The math itself is orderly and predictable. (It must be in order to generate any kind of mathematical solution, and thus make a picture.) The results are infinite. The final level of possible outcomes is so immensely huge, it seems chaotic to our limited grasp.
mytwocentsworth,fromanon-mathematician

Here's an example of chaos: Start with x = 0.3 and iterate x = 2 * x * (1 - x). The next two value of x are 0.42 and 0.4872. Iterate a few more times, and x will go to 0.5 and stay there for all subsequent iterations. Try it again with an initial x of 0.301, or any value between 0 and 1, and the final result will be 0.5. Now, try the same thing with x = 4 * x * (1 - x), just replacing the 2 with 4. If you compare the x sequence starting with 0.3 with the x sequence starting at 0.301 (or any other value close to 0.3), then you'll see that the difference between the two sequences starts out small, grows rapidly, and after maybe 20 or 30 iterations, the two sets of numbers seem to have nothing to do with each other. The first process is not chaotic, and leads to a limit point. The second is chaotic: small differences between states lead to large differences over time. In other words, there's sensitive dependence to initial conditions. This is a property of dynamical systems, not images. In the typical fractal plot for this formula, the idea of sensitive dependence on initial conditions can be seen in the fine detail around the boundary of the Mandelbrot set. One pixel may be inside the set, an adjacent pixel could be outside, and another nearby pixel could be on the boundary. The small differences in pixel location (initial condition) lead to large differences in the orbits of their corresponding points--convergent, divergent, or chaotic. In the context of math/science, a system maybe chaotic if it is governed by fixed rules. If a system's dynamics are chaotic, then they can often be visualized with fractals. Kerry

good example - i think the confusion from us non-mathemathicians come from the definition we apply to 'chaos'. we are not thinking of the word chaos in the sense of systems but in terms of true randomness. To quote one of the links i posted above: "In physics, chaos is a word with a specialized meaning, one that differs from the everyday use of the word...." "Rather, chaos refers to the issue of whether or not it is possible to make accurate long-term predictions about the behavior of the system. " cheers! GDV

I like to think of fractal images as extremely fancy graphs of equations, lol!