The Cantor Set

The Cantor Middle-Thirds Set is an example of a fractal on the real number line. This set is seen on the interval between 0 and 1 on the number line. The construction of the Cantor Set is pretty simple: (In this example, a line represents a set of numbers, and removing a section is analogous to taking out that part of the set):

Draw a horizontal line, labeling the left and right endpoints 0 and 1, respectively. This line signifies the interval of real numbers between 0 and 1.

Erase the middle third of the line (between 1/3 and 2/3). You are left with two thirds of the original line.

Erase the middle thirds from both of the new lines. Repeat this step a few times, and you will begin to see a pattern. (The pattern is actually a fractal.)

Graphic of the Cantor Set

Through infinite iteration, we will eventually end up with a set of points that remain in the Cantor Set. (They are not 'thrown out' in one of the middle thirds intervals.) Thus, a graphical representation of the set will consist of a 'dust' of scattered , unconnected points that make up the Cantor Set.

So what does this set have to do with chaos? It just so happens that certain functions, when iterated many times, produce effects that are similar to that of the Cantor Set. For instance, consider the logistic function f(x) = 4.5x(1 - x) mapped over the interval between 0 and 1 on the real number line. With every iteration of this function, the middle third of the x values are 'thrown out' of the interval [0, 1], so the effect is that only the points within the Cantor Set will stay in this interval whe n iterated through the function. (All others will escape the interval.) The concepts of staying within an interval and escaping an interval will help you to understand complex sets such as the Mandelbrot Set.