At first sight, the figures referred to by the strange term fractals seem nothing more than objects with heavily jagged contours. The nature of these intrinsically complex shapes is revealed only when, enlarging them, one tries to examine their borders more closely. Normally, when one enlarges the irregular contour of a object - especially when dealing with a mathematically defined shape - one expects to see the irregularity grow less as one keeps enlarging, until it is revealed as an overall smoothness that appeared irregular because of the scale discrepancy. None of this happens with fractals. In fact, any change of scale reveals new and surprising details that become more subtle and complex as each layer is revealed. What initially seemed to be only a jagged profile, is revealed upon inspection to be a fine, and extremely diversified structure that branches off still further with each enlargement. The irregularity of fractals is infinitely stratified. In spite of the extreme variety of shapes, the generation of many of these objects is very simple, requiring a computer program of only few lines.

Set of Julia for transformation T(z) = (1+i/2) sen z

Let's imagine a powder placed on a flat surface. Its particles can be moved on this surface, and they stay still once we stop moving them. By spreading this material differently from how its original arrangement, we operate a transformation of the plane. When the transformation has been effected, the particle that occupied the point P, will occupy a different position, indicated with T(P) to remind us that it is the transformed of P. For example, if P is identified by its cartesian coordinates (x, y), we will be able to describe the transformation T, saying what the coordinates (x1, y1) of the point T(P) are, and that they will naturally depend on the coordinates of P: Suppose now we operate the transformation T for the second time. The particle, originally in P, that was moved into T(P) will now end up in a new point T (T(P)), or rather T2(P). Transforming again, and then again, the point will keep moving into 3(P), T4(P),and so on, until, after a number of transformations, will be in Tm(P). Let us now fix a circle W with a reasonably long radius and let us ask ourselves this question: after how many transformations, will the particle originally in P, get out of W ? Obviously, the answer will depend on the initial position P of the particle. There are vast areas, starting from which, the particle will get out of W almost immediately. Other sectors of the plane, however, will be particularily enduring in the sense that we will not see the particle get out of W, within the maximum number of repetitions that we have fixed. This is the case with the simplest examples of the sets of Julia, that are obtained through transformation T, with only one fixed point of attraction, PO All the points sufficiently near PO are moved into points even closer, while all the points sufficiently far from the origin are moved into points even further away. In this situation, all the points of the plane that aren't fixed points of T, will fall into three distinct groups - those with images growing apart indefinitely, those with images growing closer to PO and the others, which have neither of these behavioural attributes, and which separate the two areas. This last set of points forms the transformation set of Julia and, because of a rather wide choice of transformations, has a fractal structure. In order to have an idea of such a structure we can operate in the following manner. Having fixed a limited region W of the plane that contains all the set of the points attracted by PO,for each point P of W, we calculate a number of repetitions sufficiently high order to get out of W, choosing for example the colour O (black) for the points that remain confined in W. As one would expect, if a point P requires N (as we shall assume) repetitions to get out of W, the points to P will require a number of repetitions near 50 (for example between 45 and 55). When, on the other hand, we move into the areas characterised by a higher and higher N we'll see that the number of repetitions varies more and more rapidly, passing from one point to another one next to it. This way, very jagged figures are formed, where the colours are merged in unpredictable, but never random, shapes. There is a method to this madness. From these constantly new and unpredictable figures the beauty of fractals is derived.