The Sierpinski Triangle
To illustrate the principle of fractals, we will create a simple (and famous) one. We start with an ordinary equilateral triangle:
Then, we subdivide it into three smaller trangles, like this:
The subdivison of the triangle into three smaller
triangles is the transformation that we are using here. The
transformation consists of three functions, one for each smaller
triangle. These three functions form an IFS. The Mathematica
source code for this IFS can be found here.
We have just applied this transformation once. This is called an iteration
of the IFS. If we iterate again, we divide each smaller
triangle into three triangles, and we end up with this:
Subsequent iterations will produce the following:
We could apply an infinite number of iterations
to our triangle, if we so desired. But, after a while, subsequent
iterations won't change the overall design that much. Even after
four iterations of this triangle, we can see what the basic
design will be, even if we iterate 4,000 times. This resulting
basic design is called the attractor of the IFS.
The attractor of an IFS remains constant even if the intial shape
changes. For example, if we start with a diamond instead of a
triangle:
and then iterate five times, we get this:
which looks very much like the image we got when we iterated the triangle five times (see above). The attractor is the same.
This attractor is known as the Sierpinski triangle. It is probably one of the two best-known fractal systems. We will explore the other one next.