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Originally Posted by Racoon
Woot! Pyrotex has the book too!?
Good....The Basic tenet is that Nature forms with a Geometrical Math pattern.
Leaf shape. Coastlines, snowflakes.....Do tell a little more Pyro....
ps. ...!!
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Well, I'm going to try to upload the executable. Maybe (
) that doesn't require having Perl. We'll see.
The fractal nature of nature is really simpler than it looks at first glance (maybe). In biology, for example, DNA doesn't have to code for the entire shape of a leaf and the twigs and the branches and the trunk. DNA only has to code for a repetitive algorithm of putting proteins here... here... there and there... and then repeating. So instead of several megabytes of storage for a description, it takes only a few hundred bytes for a cyclical sequence of operations. The real beauty of this is that each cycle does not have to be perfect or exactly like the one before. Variation adds to the beauty and complexity.
The Mandelbrot Space is of infinite complexity, but the algorith is adsurdly simple. In the region around 0, 0i in the complex plane, pick any point C.
C = a, bi. Where a and b are real numbers.
Now square C then add C -- to get D.
Now square D then add D -- to get E.
Now square E then add E -- to get F... etc.
Each time you do this, compute the distance of the point from the origin. Mandelbrot proved that if the sequence C, D, E, F, ever gets further than 2 from the origin, then the sequence will go "outward" towards infinity. Bye-bye. However, there are an infinite number of starting points (C') where the sequence C', D', E', F' always stays in the vicinity of the origin. These points make up (define) the Mandelbrot Space.
In the algorithm to "draw" the Mandelbrot Space, we count the number of times we repeat the cycle (square and add) until the distance of the resulting point exceeds 2 from the origin. We do this up to some maximum, say 512 iterations of the cycle.
If the 512th point in the sequence is still < 2 from the origin, then we color that point BLACK. These BLACK points define the Mandelbrot Space in the program. If the distance from the origin exceeds 2 at the
m-th iteration, then we color that point a color associated with the number
m. Since we typically have only k colors to choose from (say, 64) we actually use m' = mod(m, k). So if m = 68, then m' = 4 and we use the 4th color for that point.
The beauty of the Mandelbrot Space is NOT the black areas which are IN the Mandelbrot Space but the rainbow-colorful region that is ALMOST BUT NOT QUITE IN the Mandelbrot Space--the points that eventually diverge to infinity after several HUNDRED or THOUSAND iterations of the algorithm.
More later if you want. Now to see if I can upload the executable. To run it, just double-click it. -- DANG!! INVALID FILE!! Tormod may have to permit ".exe" files to upload.
Re: Fractal Geometry of Nature
