The Cantor Set was first described in the late 1800s by Henry John Stephen Smith and Georg Cantor. It has been called the first fractal, or a proto-fractal.
The generation of this set is shown in the image above. A line is taken, and presume the length of the line is 1, then the middle third of the line is removed. This leaves a linear space that is one unit long but now 2/3s in length. Recursively, this process is repeated infinitely. Each of the remaining line segments is given the same treatment, and the length of the line segments continues to decrease by 1/3 at each go. For example, here are the first few results:
- Step 1: length 1 = 1
- Step 2: length 2/3 = 0.66667
- Step 3: length 4/9 = 0.44444
- Step 4: length 8/27 = 0.29630
- Step 5: length 16/81 = 0.19753
The length of each segment is shrinking fast. Still, this is all occurring in a space that is one unit long. And, while the length of each segment is approaching zero, when calculations are completed (beyond the scope of this page) that consider what the eventual sums will be, if this is performed an infinite number of times, and that remains at 1. That is, there is no segment left in the set that has a length greater than zero, yet when we total these segments, we get the original length of 1.
Here we are seeing one of the more fascinating aspects of fractals, that our expectations based on linear thinking often don't match with the results we obtain from non-linear dynamics.
Further calculations that again are beyond the scope of this page show that the dimension of a Cantor Set is 0.63093. Recall that a dimension of 0 is a point and a dimension of 1 is a line, we see that the Cantor Set is somewhere in between, it's not a point and it's not a line.
Here is more information, at a much higher level of discussion, at Wikipedia.