Fractals and the Cantor Set

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 “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 “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