I just learned about *for* loops in Processing, and it occurred to me that they are perfect for illustrating iterated function systems. The classic example of an iterated function system is where you start with three vertices of a triangle, labeled 1, 2, and 3. Then perform the following steps:

- Plot a point (x0, y0) anywhere inside the triangle.
- Use a random number generator to generate 1, 2, or 3 randomly.
- If 1 comes up, plot the midpoint between vertex #1 and (x0, y0).
- If 2 comes up, plot the midpoint between vertex #2 and (x0, y0).
- If 3 comes up, plot the midpoint between vertex #3 and (x0, y0).
- Make your new point (x0, y0).
- Repeat the process.

Even though the locations of the points are being generated randomly, a very interesting picture emerges. Here is after 100 iterations:

And here is the result after 1000 iterations:

And here it is after 10,000. It’s pretty clear that we’re getting the Sierpinski Triangle!

Here’s the code:

float x1 = width/2;

float y1 = 0;

float x2 = 0;

float y2 = height;

float x3 = width;

float y3 = height;

float x0 = width/2;

float y0 = height/2;

size(600, 600);

background(0);

stroke(255);

frameRate(5);

for (int i = 1; i < 10000; i = i + 1) {

int roll = int (random(0, 4));

if (roll == 1) {

x0 = (x0 + x1)/2;

y0 = (y0 + y1)/2;

point(x0, y0);

} else if (roll == 2) {

x0 = (x0 + x2)/2;

y0 = (y0 + y2)/2;

point(x0, y0);

} else {

x0 = (x0 + x3)/2;

y0 = (y0 + y3)/2;

point(x0, y0);

}

}