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Living Math II – Tour De Fractal
You see them when you look at the branches of a tree. You see them in the jagged
shape of a coastline, in the branch of a fern, a river delta, and an eroded
mountain side.
Increasingly, you see them used as the cover art of new pop music albums, on the
covers of books, computer wallpapers, and if you have a geeky enough friend (term
used with utmost affection!), you might even see one on a poster of his or her
wall.
So just what am I referring to? Mathematically, their definition is fairly loose,
but the world knows them as fractals. Whoever would'a thought math could
be so beautiful?!
Fractals are unique because they bridge math and complexity in nature. The
tendency in math is to oversimplify things. But with fractals, a relatively
simple rule generates striking images that bear an uncanny resemblance to nature.
The fern fractal to the right was generated by one of these simple rules.
Interesting Aside: If you're a younger student, you
probably haven't had much experience with physics, but you will! And when
you do, you'll see what I mean about over-simplification. For example,
you will be asked to calculate how long a baseball remains in the air based
on how hard it is thrown, and how hard gravity pulls on it. But you'll
conveniently be allowed to leave out the effects of air-resistance. Why? The
original problem requires simple algebra (ok, maybe it's not simple to
you yet - keep practicing!). When you introduce complications like "air-resistance"
- the problem becomes an extremely difficult calculus problem. That is
ok for a baseball problem, as air-resistance only has a negligible effect on
a slow-moving baseball anyway. Of course, if you're going to be a scientist
at NASA and launch rockets, you'd better know your calculus - leaving out a factor like air-resistance
would cause you to miss the moon by hundreds of thousands of miles!
What IS a fractal?
Paper folding fractals,
generated with this program |
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| Image A - After a few folds... |
Image B - After a dozen or so folds |
A fractal is a geometric pattern or rule that approximately repeats itself at
smaller and smaller scales as you zoom in closer and closer to it (I said
it was a loose definition!). Look at image A: The rule that generated this
fractal is very simple. You can create one like it by taking a long strip of paper,
and folding it in half. Then take the resulting strip, and fold it in half again.
After you've folded it as far as you can, unfold it, and adjust each crease
so it's about 90 degrees (a nice right-angle) - and you'll have
something similar to image A. If you were able to fold it a dozen or so
times, it would start to look like image B.
If you don't want to actually do this with real paper, you can find a very
cool paper-folding simulation program here
to make your own.
Of course, this is a very simple example of a fractal. Fractals vary immensely
in appearance and complexity. The most common example of a more complex fractal
is called the "Mandelbrot Set," named for the French mathematician
who discovered it, Benoît B. Mandelbrot.
Fractals in Nature
As I mentioned earlier, fractals have an uncanny resemblance to many natural
phenomena. Think back to our earlier examples of fractals in nature at the beginning
of this article. Can you imagine how they all approximately repeat a pattern
as you look closer and closer? How about a tree? It starts as one "branch"
- a tree. The branch splits. Then each of those branches split, and so
on, to the little tiny twigs. If you cut a branch off at any point, it becomes
a "trunk," and you will have what looks like a little tree! 
The main thing to notice is the recurring pattern of "self-similarity"
- that is, as you zoom in, the pattern repeats itself.
This "natural appearance" phenomenon is used by 3D computer graphics programs to generate realistic looking
landscapes. Look at the program on this
page. To the right is an image from that page of a simple computer generated
landscape.
Exploring Fractals Yourself
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Basic Mandelbrot |
Zooming in... |
and in... |
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and in... |
and in... |
beautiful! |
It's very easy to explore fractals on your own. There are a number of free
programs that allow you to generate fractals of various types to your heart's
content - and of course, zoom in on them to view the self-similarity. It's
easy to get lost for hours in the alien-looking fractal "worlds."
One that I used to generate the images on the right can be found at http://www.eclectasy.com/Fractal-Explorer/index.html.
If you generate any cool fractals with the program above, send them to me at anthony@hsunlimited.com,
and we'll post them on this website along with your name.
Happy fractaling!
Interesting links mentioned in this article to free resources:
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