World of fractals
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Many people become entranced by fractals; love on first sight. They make beautiful pictures, and are a fun and enjoyable hobby. But what are their real applications? Why are fractals so interesting to mathematicians? The answer comes from their many interesting properties such as symmetry, simplicity/complexity, self-similarity, etc. They are very different from the simple lines and curves produced from most simple equations; complex patterns which are almost unpredictable unless recursively applied. Many mathematicians believed they may be used as a way of predicting complex and seemingly 'random' things.
For example, say you were walking at a constant speed, and at
every point in time you charted how far you walked. The graph
would be a straight line.
This is actually quite useful, because now the data can be
represented by the simple equation of that line, rather than the
hundreds of numbers you wrote down. It also means that you can
predict how far you will have gone at any point in the future,
because your line stretches on forever. As the situations get
more complex, an equation will help you even more.
If you were to graph the distance traveled by a free-falling ball
at short time intervals, you would get a curve, because the ball
is accelerating.
The equation for this curve is much more useful than your line
for the car. While it is not easy to compute exactly where the
ball will be three seconds from now, your curve will tell you
with a simple computation.
But now, we hit a block. Something so complex, we cannot find a
curve to match it. Graph the weather over the past ten years, and
what do you get? A seemingly random set of fluctuations that
apparently cannot be represented by an equation. This is called
chaos. There appears to be no pattern, and the only way to say
for sure where the graph will be in the future is to continue
graphing iterations, i.e., to predict tomorrow's weather
perfectly you have to wait until tomorrow! At first glance,
fractals seem the same way. They are extremely complex, and they
appear to have a random shape. But many fractals are
generated through simple mathematical equations! Is it possible
that a similar fractal equation could predict the seemingly
random weather? That possibility and other similar ones are why
so many mathematicians are studying fractals! If you could find
an equation or a set of equations that accurately matched the
weather for the past fifty years, it would almost certainly
predict the weather for the next fifty!
Unfortunately, as I said, it would predict the weather for the
next fifty years, not forever. This is because chaotic equations
(like the ones that form fractals) are very sensitive to their
input. A tiny change in a parameter can cause a huge change in
the output. Here are two graphs of a function:
The first image has the parameter for the function set at 10. The
graph, as you can see, is chaotic, because there is no pattern.
The graph on the right, however has the parameter at 10.0000001,
only one ten-millionth greater than the graph on the left. It is
also chaotic. But look! After only 40 iterations, these two
graphs are extremely different. The tiny, probably unmeasurable
difference in the input has caused a huge change in the output,
and it sure didn't take very long. That is, unfortunately, one of
the characteristics of chaotic functions. This is the reason your
weather equations would eventually fall apart. Unless you could
get the exact equation for the weather (which would be
impossible), it would eventually stray from the correct
predictions, even if it was super-close to the perfect equation.
Therefore, in a few decades, or even in a few years, a new set of
equations would need to be found.
You may look at the universe and say that everything is placed
randomly. All the galaxies, planets, and stars are random, as far
as we can tell. Even our weather seems to be completely random.
But many mathematicians no longer believe in "random."
More and more they are starting to believe that there is a
chaotic equation to describe any apparent randomness. Fractals
may be just be something more to add to our descriptive shapes.
Instead of "there's a squiggly line and a blob over
there," we might be able to say, "there's a
fractal." We might be able to describe organic materials
much more accurately by using fractals and chaotic patterns than
by using curves and lines. And we may be able to use fractals as
additional types of equations to which we can map our data.
Fractals and Chaos are relatively new branches of math, since
they cannot be explored without powerful computers invented only
recently. Without a doubt they have already improved our
precision in describing or classifying "random" or
organic things, but maybe they are not perfect. Maybe they are
just closer to our natural world, not the same. Then there
are those who believe that true randomness does exist, and no
mathematical equation will ever describe it perfectly. So far,
there is no way to say who is right and who is wrong. This is
something that you can decide for yourself as you explore the
infinite worlds of fractals.
If you'd like to learn more about the applications of chaos theory, visit the pbourk fractal site here.
Special thanks to The Fractory for the pictures and the basis for these lessons.
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