Fractals were theorized and studied by Benoit B. Mandelbrot (Warsaw, 1924, living) in its successful book "The fractal geometry of Nature" (1982) that puts the basis of geometry and mathematics about fractals and described the inability of Euclidean geometry to describe complex forms in nature.

The simplest definition of a fractal describes it as a geometrical figure in which an identical motif is repeated up to an infinitely reduced scale, so that, magnifying the figure, recurrent shapes can be seen smaller and smaller, up to infinite.

So, differently from other geometrical figures, a fractal shows more particulars every time it is magnified.

The word "fractal" was created by Mandelbrot himself who derived it from the Latin "fractus" that means "broken, fractured" to name this geometrical process in which a straight line is "fractured" in a broken line of infinite segments.

Mathematically, the fractal dimension is not a whole; the length of a plain fractal can't be measured definitely but it strictly depends on the number of iterations made on the initial figure.

We can consider, for example, the most studied classic fractal: the C set of Cantor.
This starts from a segment [0,1]. This group is formed by points remaining on the segment after that we have cut away its central third (1/3, 2/3) at the 1st iteration (p = 1); then, we do the same from each one of the 2 resulting segments [0, 1/3] and [2/3, 1] at the 2nd iteration (p = 2) and so on for every segment remained, to infinite.
Clearly, for p -> INF. , the C set is formed by the extremities of shorter and shorter segments formed at every iteration, i.e., infinite points.
The length L(C) of cut away segments after the p(n) iteration is given by the expression:

L(C) = 1/3 + 2(1/3^2) + 2^2(1/3^3) + .... = 1/3 x SUM[(2/3)^k],
from k = 0 to k = p.
Obviously, L(C) = 1 for p -> INF.

So, it's showed that the overall length of Cantor set is = 0, but also that is formed by infinite points! We must realize that the classic definition of dimension in this situation is totally ineffective. A point, in fact, has no dimensions and its length is = 0 so, a sum of infinite zeroes is always zero.

Another example is the Van Koch curve, without tangents in every point.
It starts from a segment to which we cut away the central third replacing it with 2 consecutive segments, each one equal to that cut away (p = 1), so that we have now 4 segments.
We repeat this operation on each of these 4 segments and the result (p = 2) is a broken line of 16 segments. We keep on doing the same on every segment up to infinite.
The length of this curve is (4/3)^p, tending to infinite for p -> INF..

So, we never could know what could be the final shape of a fractal, but we are always limited by an approximation, as good as we want, but not the exact fractal
Just like trying to define exactly an irrational number.

The same situation in the so-called "not linear systems"; it's not possible determine the final situation given only the starting conditions, but always new experimental data must be added.
These topics have started the complex argument of "deterministic chaos" of inordinate situations.
We can't know the final shape of a system with infinite iterations, but we would know well how to calculate it.

Fractals have been used to describe natural shapes, like the development of a fern leaf, whose details reproduce always the same figure or the growth of a snow-flake and simply to create beautiful images using various colours.

Learn more about this author, Aldo Bonincontro.
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