One of the interesting aspects of fractals is how easy and quickly one can make beautiful designs. Use this Sierpinski fractal applet and select different color combinations as you move from one iteration to the next. Reset it and try different combinations. You can also select the random color option and let it randomly select a new color for every iteration. Reset it, move up several iterations, and see how quickly you can produce colorful geometrical patterns.

How does the Sierpinski Triangle applet work? Reset the fractal and click the right arrow button to go to the first iteration. As you can see, it starts with an equilateral triangle (the axiom) which gets divided into four identical triangles by inserting one triangle which is pointing down (the production rule).  The area of the triangle that gets inserted is one fourth the area of the original triangle and it gets inserted right in the center. To go to the next iteration, the exact same process is done recursively to each of the resulting triangles that are pointing up. Very simple!

How is the Sierpinski Triangle applet implemented? This applet uses geometrical mathematics to calculate the vertices of the new triangle(s) to be inserted. It makes and maintains a list of triangles which grows pretty fast with each iteration. Using this method it is very easy to assign colors to each triangle. Could this fractal have been done using the L-system method? Keep reading and you'll find out.

How quickly do the number of triangles grow? Lets do one math exercise regarding this issue. By re-running the applet you can easily fill in a table which includes the numbers of triangles pointing down. Here is an example of such a table: 

Can you come up with a general formula to predict the number of triangles for any iteration? To figure this out you probably need to look at both the number of triangles pointing down and the number of triangles pointing up.