Fractals/Cantor's Set

Cantor's set is a subset of an interval in the real line - normally the unit interval, [0,1]. It is constructed by removing the middle third of the interval but leaving the end-points where they are (see open intervals). This iterative step is then applied again and again to the parts left "to infinity" to any and all line segments left. This eventually (after an infinite length of time and number of iterations) reduces the original set to a set of distinct points. The picture on the right shows the source set

Taking a probability measure over the line (analogous to determining how much of the length of the original line is left at any given time) one can see that one loses 1/3 per iteration multiplicatively. Hence, the length left decreases exponentially to zero with the number of iterations increasing. The limiting behaviour of this measure can be described by examining the function $$f(n)=\left ( \frac{2}{3} \right )^n$$, noting that its limit lies at zero. In fact, the cantor set, the limiting residue of the line after an infinite number of iterations as described above is a set of distinct points. Since the points have no length, this is described as a set of measure zero. It is also an infinite set, as 2^n more endpoints are created with every successive step and these end-points are all that is left within the set. It is also an uncountable set, which means that the natural numbers ("whole" or "counting" numbers) can't be singularly mapped onto it by a process that allows them to be mapped back.