## FRACTALS

Euclid’s geometry, besides being a powerful deductive instrument, has been extremely useful in many fields of knowledge. But it has also constituted a considerable abstraction of reality, by considering ideal situations as normal.

When we propose to calculate the length of a natural figure, the problem arises if we want to consider the details, because we are faced with the difficulty of measuring all the breaks and folds of the object to be calculated. Lines and curves are generated in a process of repetition that adds more detail at each step, extending its length to infinity.

Differentiable geometry provides models suitable for a certain form of regularity that allows us to approximate the most complex geometric shapes by means of simpler ones: straight lines, planes, etc, but in Nature these figures are the exception, while most of the shapes around us are fractal. Fractal Geometry offers an alternative model that seeks a regularity in the relationships between an object and its parts at different scales; it studies the geometric aspects that are invariant with the change of scale.

Unlike differentiable geometry, where elements can be generated directly, in Fractal Geometry the shapes or objects are sets of mathematical procedures or algorithms that rotate, translate, rescale or deform in a particular way; in this way fractal geometry is constituted by an infinity of elements, where each complete transformation represents a unique figure.

Fractal geometry has turned out to be the one that best describes most of the objects around us, which are the product of complex dynamic processes, contrary to the idea that was held for centuries that the most common shapes were those described by Euclidean geometry, which obey a linear, static and predefined mathematics.

*"Clouds are not spheres, mountains are not cones, coastlines are not circles, and tree barks are not smooth, nor does lightning travel in a straight line."*

## BENOIT MANDELBROT

Mathematician Mandelbrot coined the term fractal (from the Latin fractus, broken, ruptured) in the mid-1970s to designate certain irregular geometric objects that are present in many behaviors and shapes in nature. With them he constructed a set of new rules, to explore the geometry of nature. In Mandelbrot’s own words: «Nature demands not only a higher degree but also a different level of complexity».

But the origin of fractals dates back to before Mandelbrot and goes back to the end of the 19th century, with the «Cantor Set» and the theories of Henri Poincaré. At the beginning of the 20th century. Gaston Julia and Pierre Fatou, had developed their idea of making interactions with complex numbers, and it is above all the results of these last works that served as a basis for Benoit Mandelbrot, so that years later he presented us with the discovery of Fractal Geometry.

## PROPERTIES

Fractals are geometric sets that originate through the continuous repetition of a specific mathematical process. This process, which is usually of a very simple nature, is the determinant of fractal structure, complicated in appearance but requiring very little data for their description and construction. It is in the possibility of construction and mastery

of very complex structures through very simple processes that lies the power of fractal structures to model and explore the phenomena of Nature.

The most notable properties of fractal structures and objects are: self-similarity at any scale, which consists in the similarity of any part with the whole (the unity in multiplicity or the correspondence principle), the iteration process to obtain them, and the fractal dimension, which is usually determined by irrational numbers, so they are described in several dimensions at the same time and do not correspond totally to any of them. Other peculiar characteristics that these objects usually have are to have defined areas but with infinite perimeters and to have detail at any scale of observation as well as being defined by simple algorithms.

Fractal objects are figures that do not correspond to a single dimensionality. As we approach the object we observe that its dimension will change, everything will depend on the scale at which the observation is made. The degree of irregularity of a fractal will be the same as we change scale, which is a paradox, since it will be a regular irregularity.

Thus the fractal dimension of an object will be determined by its fractional values, so these regions or dimensions will be defined by the parameters of the system. In this way its geometric dimension will always be between line and surface, or between surface and volume, or between volume and time… in n-dimensional spaces.

However, there are many systems in which it is not only one, but several parameters that make it up, so that distinguishing the boundary between periodic and chaotic behavior is not easy. Although the behavior of a system may be regular and orderly in a given time interval, it can and does happen that it suddenly becomes chaotic and shows irregular characteristics.

These nonlinear systems show a behavior under certain unpredictable conditions, in spite of not having any influence of chance and being entirely deterministic. In them one or several parameters can be identified on which their behavior depends. In these cases the behavior will be stable or chaotic or even both, depending on the values of the parameters.

## THE GEOMETRY OF MOVEMENT

We have called «Chaos» to everything that we cannot systematize, and we often relate disorder with randomness, but what may seem chaotic or random to us, from another distance or another point of view will seem orderly or that somehow responds to a certain scheme or pattern.

The ideas developed from fractals and the Chaos Theory are of great importance not only conceptually but also in terms of practical application. In recent years, new ideas have emerged that are very useful for describing and understanding the multitude of phenomena that occur in various branches of knowledge. Fractal Geometry has become a basic tool in different fields of science and the use of fractals and Chaos Theory is fundamental in the description of complex phenomena.

In addition to all these new scientific conceptions generated from this «Chaos Geometry», it has caused a true revolution in all fields, not only mathematical or scientific, but also artistic, philosophical and of thought in general.

A new geometry that allows us to reproduce and model most of the forms and dynamics of the natural systems and processes that surround us; a true Geometry of Nature that provides us with a new understanding of Chaos.