# What is fractal?

## What Does fractal Mean

Mathematical expert Benoît Mandelbrot 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 part of the dictionary of the Royal Spanish Academy (RAE) .

A fractal is a figure , which can be spatial or flat, made up of infinite components . Its main characteristic is that its appearance and the way in which it is statistically distributed does not vary even when the scale used in the observation is modified.

Fractals are, therefore, elements classified as semi-geometric (due to their irregularity they do not belong to traditional geometry ) that have an essential structure that is repeated at different scales.
The fractal can be created by man, even with artistic intentions, although there are also natural structures that are fractals (such as snowflakes).
According to Mandelbrot , fractals can present 3 different classes of self - similarity , which means that the parts have the same structure as the whole :
* exact self-similarity , the fractal is identical at any scale;

* quasi-self-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 conserved with the variation of the scale.
Fractal techniques are used, for example, to compress data . Through the collage theorem , it is possible to find an IFS (Iterated Function System), which includes the alterations that a complete figure undergoes in each of its self-similar fragments. As the information is encoded in the IFS, it is possible to process the image.
We speak of fractal music when a sound is generated and repeated according to patterns of spontaneous behavior that are very frequently found 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 this is not correct. Its definition, and which usually generates such confusion, is the following: a segment is taken and divided into three, to then eliminate the central one and repeat said action infinitely with the rest.
The fractal dimension
Classical geometry is not broad enough to cover the concepts necessary to measure the different fractal shapes. If we take into account that these are elements whose size changes incessantly, it is not easy, for example, to calculate their length. The reason is that if you try to measure 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 on the right, it can be seen that from its birth it grows a third along each step; in other words, the length of the portion that is located at the beginning increases without end, determining that each curve is 4/3 of the preceding one.
Since the length of the fractal line and that of the measuring instrument or the chosen unit of measurement are directly related, it is absurd to use this notion. That is why the concept of fractal dimension has been created that allows, when we speak of fractal lines, to know in what way or to what degree they occupy a portion of the plane .
In relation to traditional geometry, a segment has dimension one, a circle, two, and a sphere, three. Since a fractal line does not span the entire plane portion, it should have a dimension less than two.

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