Fractals and the Cantor Set :: Fractal Cantor Essays

Fractals and the Cantor Set Fractals are remarkable designs noted for their infinite self-similarity. This means that small parts of the fractal contain all of the information of the entire fractal, no matter how small the viewing window on the fractal is. This contrasts for example, with most functions, which tend to look like straight lines when examined closely. The Cantor Set is an intriguing example of a fractal. The Cantor set is formed by removing the middle third of a line segment. Then the middle third of the new line segments are removed. This is repeated an infinite number of times. In the end, we are left with a set of scattered points. These points have some very curious properties. First, there are an infinite number of them. In fact, there are so many points that no matter what list we create or what rule we apply, not all of the points will appear, even if our list is infinite. In other words, the set belongs to aleph-one. This is demonstrated through diagonalization. Here’s how—first one endpoint of the original line segment is labeled zero. The other endpoint becomes one. All the points in between are assigned fractional values. We can calculate more easily if we assign the values in tertiary, the base-three system. Unlike the common decimal system, the natural numbers are labeled 1, 2, 10, 11, 12, 20, 21, 22, 100, and so forth. Notice that the places of the digits represent the powers of three rather than the powers of ten. The â€Å"decimal places† represent 1/3, 1/9, 1/27, and so forth. The first removal takes out all points between .1 and .2. The second removal takes out all points between .01 and .02 as well as the values from .21 and .22. By continuing these specifications, all numbers that contain a â€Å"1† are removed, (except numbers ending in a one, such as .220021) and number containing merely twos and zeros are kept. The numbers ending in 1 are re-written by replacing the final 1 with 02222222222222†¦. because this is equal to 1 in tertiary. Suppose that we could somehow count all Cantor Set elements in one list. Then we could write out that list in order, one above the other. However, if we took the first decimal of the