Famous Fractals
Cantor Set
Cantor Set is one of the most famous fractal of all, yet it is the most simple one. It is an example of a fractal dust, which is formed by cutting parts of a line segment.
DUST METHOD
Cantor Set, as a dust, can be formed by repeatedly cutting out middle thirds of a line segment:
GENERATOR ITERATION
The same generation method can be looked at as generator iteration with the following base and motif:
Thus, we can consider the Cantor Set to be a base-motif fractal.
FRACTAL DIMENSION
The Cantor Set is composed of 2 identical shapes, each of which is 1/3 the size of the entire figure. Using the similarity method, the fractal dimension is log 2 / log 3, or approximately 0.63.
LENGTH
By cutting out the middle third, we are making the length of the figure 2/3 its previous length. After an infinite number of iterations, the length will become 0. Indeed, the Cantor Set is a collection of disconnected points.
The Cantor Set, being a collection of points is very hard to show. We can give it height and use a bar instead of a line segment, which will give us a Cantor Bar. We can also present the fractal using other fractals, such as the Cantor Comb, Cantor Curtains, Cantor Square, and the Devil’s Staircase, which are 2-dimensional and can be easily seen.
The pattern of the Cantor Set was found in the rings of Saturn and in the spectra of some molecules.
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