Where does THE formula come from ?

[bib]

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=z²+c ?

Any information on the genesis of the formula much appreciated.

Thanks

[David Makin]

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.

[HPDZ]

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.

[cKleinhuis]

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!

[gamma]

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!

Oh dear...

[cKleinhuis]

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/