   
Fractals, in Layman's Terms
  

   
This is a simple question with a very complicated
(and very long) answer. A technical answer,
while accurate, doesn't help much because it uses
other fractalspeak jargon that few people understand,
so I won't even give that definition here.
The simple answer is that a fractal is a shape that,
when you look at a small part of it, has a similar
(but not necessarily identical) appearance to the
full shape. Take, for example, a rocky mountain.
From a distance, you can see how rocky it is; up
close, the surface is very similar. Little rocks
have a similar bumpy surface to big rocks and to
the overall mountain.
  
This concept of selfsimilarity can be a little hard
to come to grips with, but it's a fundamental part of
fractals. So take a look at this first image—from
a Julia set, a very simple fractal type. I've
highlighted a small box near the left side (it's a little
faint). The portion of the image in that box is shown
in the second image, below. That image also has a small
box highlighted, and that area is shown in full in the
third image.
You can see in these images that smaller areas of the
fractal shape look very much like the larger, fullsize
image. With Julia fractals, you can continue this
enlarging ("zooming") process as often as you like, and you will
still see the same sort of details and shapes at very
tiny sizes that you see on the fullsize image. This
is what is meant by selfsimilarity.
Now of course, with something so rigidly selfsimilar,
there's not really much point in zooming in. After all,
everything is the same; small detail looks like large
detail. So while it's interesting that fractals
are selfsimilar, if this is all there is to it, there
isn't much point.
Fortunately for fractal enthusiasts, that isn't
all there is to it. Many fractal types get wildly
different as you zoom in. They're still selfsimilar,
but they're not rigidly selfsimilar.
This is what makes fractal exploration so intriguing. The
features you see as you zoom are always changing—teasing
you with a little bit of familiarity, and tantalizing
you with new and unexpected twists. With just a single
fractal shape, you can explore forever and never
see everything it has to offer. The further you zoom, the
more likely you are seeing something that nobody has ever
seen before. And with modern computers, it's very easy to
zoom and zoom and zoom. With just a few clicks you can have
zoomed so far that the original fractal image is larger
than the sun.
  
Take, for example, the zoom sequence on the left.
Starting with the upper left image, the center of each
image was magnified tenfold. Whereas the Julia image
looked virtually identical as it was zoomed, this
Mandelbrot image shows new variations as the magnification
increases. (Because of the small size of these images, it
may not be obvious that each image is a magnification of
the previous one, but they are.)
The basic technique of these fractals can actually
be explained without resorting to confusing
mathematical equations and jargon. It's rather
simple, really.
First, give every point on the screen a unique
number. Now take that number and stick it into
a formula; you'll get a result from the formula.
Take that result and stick it back into the
formula. Keep doing this and watch what
happens to the numbers you get. Color each
point based on what happens.
That's it. Really—that's it. Now, with
most formulas it probably won't do much of interest,
but with the formulas used in fractal creation,
some interesting things happen. Sometimes the
numbers you get by feeding the results of a
formula back into the formula (iterating)
explode into enormous numbers, that just
keep getting bigger and bigger. Those points
get colored one way. Other times, the numbers
"home in" on a number, getting closer and closer
to it. They get colored a different way.
The interesting thing—and the reason fractals
work at all—is that sometimes, just a tiny
little change in the number you start with can
completely change what happens as you keep
iterating the number. And the boundary
between numbers that explode and numbers that
home in is complicated and twisted—it's the
shape of the fractal.
Calculating fractals this way involves a lot
of work. A small fractal image—perhaps
only 640x480—contains over 300,000
points. Each of those points may require running
a number through the fractal formula more than
1,000 times. This means the formula has to be
computed more than three hundred million times.
And that's a mild example. Extreme images
(such as postersize fractals) can involve
more than one trillion calculations.
Fortunately for the impatient among us, modern
computers are fast enough to do the job in a few
minutes. Large fractals might take hours or days,
but exploring fractals has never been easier.
As I stated at the outset, fractals are a huge
topic. All I've even talked about here are one
particular type of fractals (escapetime
fractals), but there are many other types as well.
Unfortunately, the further you look into fractals,
the more math you will need to know. There are very
few fractalrelated books or web pages that don't
get into heavy mathematics. I've attempted to assemble
some pages here that will get you started on the
mathematics behind fractals in an accessible fashion,
but there is no hiding the fact that it is
math.
 
