Fractals and the Cantor Set :: Fractal Cantor Essays

Fractals and the hazan roachFractals are rare designs noted for their infinite self-similarity. This means that sm in all parts of the fractal contain all of the information of the entire fractal, no consequence how small the viewing windowpane on the fractal is. This contrasts for example, with most functions, which tend to look like straight births when examined closely. The Cantor Set is an intriguing example of a fractal.The Cantor set is organise by removing the lay third of a line segment. Then the middle third of the new line segments are removed. This is repeated an infinite scrap of times. In the end, we are left with a set of scattered points. These points nurse some very curious properties.First, there are an infinite keep down 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 go out appear, even if our list is infinite. In other words, the set belongs to aleph-one. This is demons trated through with(predicate) diagonalization. Heres howfirst one endpoint of the original line segment is labeled zero. The other endpoint becomes one. All the points in amidst are assigned fractional value. We can calculate more intimately if we assign the values in tertiary, the base- triple system. Unlike the common decimal system, the inseparable 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 decimal places represent 1/3, 1/9, 1/27, and so forth. The first remotion 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 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 fina l 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