Now imagine you take another complex number, z. You will run this
number through this iterative equation:
What this means is that from any z you will figure a new z based on multiplying the old z by itself and adding the constant value, c. As you keep doing this, you'll end up with a whole sequence of z values. There are three possibilities for this sequence of z values:
1. the values increase without bound (towards infinity)
2. the values collapse (to zero)
3. the values change, but do not seem be (1) or (2)In fractalspeak, infinity and zero are called "attractors" because lots of points end up heading towards (are attracted by) these places. All the points that fall into class (3) are parts of the "Julia Set".
The question is, how does a computer know which points are which? There is
a way in which a computer can generate only points in class (3), but it's
sort of random, so you can't guarantee that you'll get a complete picture
of the set after any given amount of time. So basically what happens is
the computer maps each point on the screen to a complex number z, and tries
every point to see whether it's type (1), (2), or (3). Most programs
that do this treat (2) and (3) the same; this is called the "inside" of the
set and I usually (but not always) color it black. "Outside" points are those
which fall into class (1); they are usually colored based on how "quickly"
the computer can determine they're in that class, which results in the
pretty colored bands. There are lots of other ways to color the images,
but the basic shape - the outline of the two areas, "inside" and "outside" - is
determined by this relatively simple mathematical method.
As you might imagine, trying every single point on the screen to see which
type of sequence of z values it makes takes a long time. A 1024x768 screen
(the size I do most of my exploring at) has over 750,000 points. Some of
those points may go 1,000 or more times through the iterative equation before
the computer gives up. It can often take hours to make a Julia set picture,
and sometimes (like when I made poster size images, 10240x7680) it can take
days.
