Heed some beginner's help

[mooserider]

Hi all, this is my first post here. I'm having a heck of a time figuring out a few things and was looking for some help. And to be honest, I'm not even sure 'fractals' are what I'm looking to build. I get a bit comfused on the difference between fractals and 'chaos' (or rather, the images that can be produced through chaos). So please forgive me, as I'm bound to use incorrect terminology and probably won't "get" some of what you guys have to say.

Ok, so what I'm working on is writing some little graphics routines that will create pictures from 'chaos' algorithms. The first book on chaos that I read (Chaos: Making a New Science) described a fascinating thing... how you can take one of little chaos toys, with the pendulumn and the magnets, and graph out the end results of where the magnet stops to produce a picture. Another example given was taking a cross-section of the path of an orbiting body (in 3D)... and how the image produced there also.

So one question is, are these things considered 'fractals'? Every time I look up info on how to produce a fractal, all I see are formulas. It seems there are two ways to produce them... one being the use of some chaotic algorithm that mimics nature, and the other through formulas (I'm assuming that they're really one in the same, but I just don't understand enough about them yet). I can't understand the formulas, but I can easily map one out with the natural method (i.e., mimic some natural phenomenon such as the chaos toy).

What I really want to know is where to find out more about the production of fractals using natural means like the chaos toy. Is there a term for this? What do you call the production of a fractal via the chaos toy? I would like to look up more info on it, but I'm having a hell of a time doing so because I don't know what to search for.

Keep in mind that I don't have a ton of math knowledge. I'm familiar with only the basics of calculus. And I have no idea what sort of math you need to understand the formulas I typically see for fractals. That's basically why I'm using the method I am for producing them... because you need relatively no math skills to do it.

Thanks for any help in advance! Sorry it's such a long post... but it's hard to describe something for which you don't have a name.

[lkmitch]

Fractals and chaos are related, but different.  You mention the magnetic pendulum (an excellent example of chaos); think of the standard, boring pendulum, without friction.  Once started, it just swings back and forth, always rising to the same height.  If you make a graph of the angle of the pendulum bob on one axis and its velocity on the other, you'll wind up with an oval (an ellipse).  Without friction, the graph will continue drawing the same curve.  (With friction, the graph will be a spiral, eventually ending in the center when the bob has zero velocity and zero angle.)  This graph is a picture of the pendulum's "phase space," and relates an ellipse to the smooth dynamics of the pendulum.  In this sense, fractals illustrate the dynamics of chaos:  the graph of chaotic system dynamics in phase space is often a fractal.  So, to generate fractals from chaos, one way is to plot the dynamics is phase space and to take a section of that graph.  Another approach would be to plot the position of the moving thing at a particular point in time as a function of system parameters.  This is what typical Newton's method fractals often do:  the final value of the root (color) is plotted at pixel positions representing the initial value of the variable.

For more information on these ideas, look up Lorenz's butterfly or the Henon map.

Kerry

[cKleinhuis]

hi there,
the pendulum is indeed a good point to start if you are interested in chaos theory,
there was a great video explaining the basic of fractals and also mentions very good how easily chaos is obtained using a simple formula!

http://www.fractalforums.com/fractal-related-links/the-mandelbrot-set/