Fractals and Chaos
Sure, fractals are often beautiful and intricate, and of course they make wonderful backgrounds
for your desktop. But what are they really good for? What makes fractals so interesting to
scientists and mathematicians? The answer is the fact that they are both very simple and infinitely
complex at the same time. They are not the boring straight line or simple curve we come to
expect from such a simple equation, instead they are complex patterns, and many scientists believe
that they may be useful as a way of predicting apparently unpredictable or random things.
Take for example a car traveling at a constant speed. Say you were to write down the position of the
car at every point in time, and then graph that data. Your graph would be a straight line.
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 the
car 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.