Fractal

Lesson

Cantor set

The first fractal object that we studied was an infinitely long curve and with dimension larger than 1. We define an almost empty set, consisting of a self-similar distribution of singularities.

We also build the object by successive iterations even starting from a length of a segment that we will divide into three equal parts, but here the result of the first iteration will be obtained simply by removing the central segment.

The generation of the Smith-Volterra-Cantor set. Every step, remove the central 1/22n from each bar. The top bar here is Step 0, the bottom is Step 5

We then obtain a disjoint set formed of two segments of length separated by a gap of the same length. The construction will be continued indefinitely, with each iteration by dividing each segment into three present and amputating its central third. The length of initial segment was equal to . The total length of two segments obtained at the end of the first iteration will be equal to and the total length of the segments obtained at the end of the nth iteration is equal to:

$L_n = a \Big( \frac{2}{3} \Big)^n$

If we apply the technique of overlapping already mentioned above for calculating the fractal dimension of this object there is need to times of overlaps when , so :

$2 = 3^{\Delta} \quad and \quad \Delta = \frac{\log 2}{\log 3} = 0.6309$

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