Title: Where does THE formula come from ? Post by: bib on January 25, 2009, 06:56:20 PM Hello,
The are quantities of fractal web sites out there, but I was not able to find the answer to this apparently "simple" question : How B. Mandelbrot came out with this magic formula : Z=zē+c ? Any information on the genesis of the formula much appreciated. Thanks Title: Re: Where does THE formula come from ? Post by: David Makin on January 25, 2009, 09:29:41 PM I'm guessing that the answer is straightforward - simply because z^2+c is basically the simplest non-linear (and non-trivial) rational function that could be used for the further investigation of Julia's research on the iteration of rational functions.
Title: Re: Where does THE formula come from ? Post by: HPDZ on January 26, 2009, 04:54:32 AM Agreed. A quadratic map like that is the simplest thing that gives the two necessary elements to have chaotic behavior -- folding and stretching. Folding means that once you do the operation, the inverse has two possible values (i.e. x^2=1 could mean x=1 or x=-1). Stretching means that the mapping is not linear, as David pointed out.
It's amazing still that such complexity comes out of something so simple. And so universal, once you've got the computing power. I wonder if when aliens find the Voyager probe they think we were a backwards civilization because we didn't have a Mandelbrot set etched into the thing. Title: Re: Where does THE formula come from ? Post by: cKleinhuis on January 26, 2009, 11:17:28 AM I wonder if when aliens find the Voyager probe they think we were a backwards civilization because we didn't have a Mandelbrot set etched into the thing. this really is a shame! :angel1: Title: Re: Where does THE formula come from ? Post by: gamma on January 26, 2009, 07:05:29 PM I wonder if when aliens find the Voyager probe they think we were a backwards civilization because we didn't have a Mandelbrot set etched into the thing. this really is a shame! :angel1: Oh dear... Title: Re: Where does THE formula come from ? Post by: cKleinhuis on January 26, 2009, 07:12:10 PM if you are interested in the source of chaos, read the very good article on wikipedia on logistic function : http://en.wikipedia.org/wiki/Logistic_function and the following section contains an interesting video tutorial on mandelbrot equation: http://www.fractalforums.com/introduction-to-fractals/the-mandelbrot-set/ O0 O0 O0 O0 |