The math expert Benoît Mandelbrot He was responsible for developing, in 1975 , the concept of fractal, which comes from the Latin word fractus (can be translated as "broken" ). The term coined by the French was soon accepted by the scientific community and is even now part of the dictionary of the Royal Spanish Academy (RAE) .
A fractal is a figure , which can be spatial or flat, consisting of components infinite. Its main characteristic is that its appearance and the way in which it is distributed statistically does not vary even when the scale used in the observation is modified.
Fractals are, therefore, qualified elements such as semi geometric (due to their irregularity they do not belong to the geometry traditional) that have an essential structure that is repeated at different scales.
The fractal can be created by man, even with artistic intentions, although they also exist structures natural that are fractals (like snowflakes).
According to Mandelbrot , fractals can present 3 different kinds of self-similarity , which means that the parts have the same structure as the set total:
* exact self-similarity, the fractal is identical to any scale;
* Quasi-similarity, with the change of scale, the copies of the set are very similar, but not identical;
* statistical self-similarity, the fractal must have statistical or number dimensions that are preserved with the variation of the scale.
Fractal techniques are used, for example, to compress data . Through collage theorem , it is possible to find a IFS (system of iterated functions), which includes the alterations that a figure complete in each of its self-similar fragments. When the information encoded in the IFS remains, it is possible to process the image.
We talk about music fractal when a sound is generated and repeated according to patterns of spontaneous behavior that are found very frequently in nature. It should be mentioned that there are computer programs capable of creating compositions of this type without human intervention.
The Cantor set is often cited in relation to fractals, although it is not correct. Its definition, and that usually generates such confusion, is as follows: a segment and it is divided into three, and then eliminate the control panel and repeat said action infinitely with the rest.
The fractal dimension
Classical geometry is not broad enough to encompass the concepts necessary to measure different fractal forms. If we consider that they are about elements whose size changes incessantly It is not easy, for example, to calculate its length. The reason is that if you try to make a measurement of a fractal line using a traditional unit, there will always be components so small and thin that they cannot be precisely delimited.
In the Koch curve, graphed to the right, it can be seen that from one birth a third grows along each step; in other words, the length of the portion that is located at the beginning is increased endlessly, determining that each curve is 4/3 of the previous one.
Since the length of the fractal line and that of the measuring instrument or the chosen unit of measure are directly related, it is absurd to use that notion. That is why the concept of fractal dimension has been created that allows, when we talk about fractal lines, to know how or to what extent they occupy a portion of flat.
In relation to traditional geometry, a segment has dimension one, a circle, two, and a sphere, three. Since a fractal line does not cover the entire plane portion, it should have a dimension that does not reach two.