Fractals – Origin of The Fractal Geometry.

In 1623, Galileo Galilei expressed that the "language" of nature are the Mathematics, with Euclidean Geometry as one of its "dialects" and, that the "alphabet" of that dialect is formed by traditional geometric figures.

But in the next 400 years, the development of Science showed that Galileo did not quite hit on the "dialect" and "alphabet" of the "language" of the Universe. In this regard, let us see the following sentence of the Polish-French mathematician Benoît Mandelbrot (1924-2.010), given in his book "The Fractal Geometry of Nature":

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line… Nature exhibits not simply a higher degree but an altogether different level of complexity.”

^{Figure 2: Benoît Mandelbrot (1.924-2.010) (CC0 Creative Commons).}

Mandelbrot affirms, that the classic geometric figures are not the most adequate to represent structures and phenomena of nature, like the branch of a tree or a bush like a fern, the coast of an island or a continent, the profile of a lightning bolt, the movement of pollen particles in water or air (the well-known Brownian movement), among many others, since classical figures tend to lose their shape when they are amplified, that is, when they are "zoomed" , situation that does not happen in the natural world.

For example, of a branch of a tree leaves many branches and, in each of them, the same pattern is repeated, that is, a "zoom" or extension of a part of the original branch is very similar to the original branch. The branches of a tree or lightning have irregular shapes, but when they are enlarged their forms are repeated.

Mandelbrot, proposes that the "ideal" geometric figures to represent natural bodies, must contain "copies" of themselves or "similar" copies in each of their parts and that these can be observed through successive enlargements, as occurs in nature itself.

Such figures, which in addition to the property described, have other characteristics, were called fractals by Mandelbrot, in 1975, although many had been proposed and studied from almost a century before, but in isolation and without having been found common points in them.

^{Figure 3: Similarity between a tree and its branches (CC0 Creative Commons).}

Given the complexity of the forms of the world, Mandelbrot proposes that a more general and appropriate "dialect" constitutes it Fractal Geometry, being The Fractals, the characters of its "alphabet".

Mandelbrot, is considered the father of Fractal Geometry, not so much for having created it, since most of the principles of this multidisciplinary branch of Mathematics and Physics already existed, but for the fact of amalgamating them in a single and coherent discipline, and of course, to give his name.

DEFINITION AND CHARACTERISTICS OF A FRACTAL

Mandelbrot, defined a fractal as:

A set whose fractal dimension is strictly greater than its topological dimension.

^{Benoît Mandelbrot.[1]}

Fractal dimension? Topological dimension? In a next post i will talk about these dimensions!

But, the same Mandelbrot, suggested that perhaps it was better to leave the word fractal without a "limiting" definition, a situation similar to the one that occurred with the definition of "life".

However, two of the most influential researchers in the area^[2][3], define a fractal as a set that has the following characteristics:

• It has a fine structure, that is, details on all scales.
• It is irregular as to be described in the traditional geometric language.
• Has some form of self-similarity, total, approximate or statistical, that is, its parts are similar to the whole.
• Generally, its fractal dimension is fractional and greater than its topological dimension.
• It is constructed by means of a recursive process or iterative.

^{Figure 4: Self-similarity in the Mandelbrot Set.}

Mandelbrot states in his book "The Fractal Geometry of Nature" that he was forced to "create" the word fractal to identify objects whose dimension was fractional, since there was no one that referred to them. He explains that the "coined", from the Latin adjective fractus, participle of the Latin verb frangere, which means breaking into pieces, creating irregular fragments.

Affirms, that the word "is reasonable, and it comes from pearls!"^[1], because it evokes the word fraction, and as we saw fractals have fractional dimension, and also, it also means irregular, behavior of many fractals, such as the curve of the Brownian movement.

REFERENCES

^{[1] Mandelbrot, B. (1983). The fractal geometry of nature. Freeman and Co., New York.
[2] Falconer, K. (1990). Fractal Geometry: Mathematical foundations and applications. Jhon Wiley & Sons Ltd., New York.}

^{[3] Barnsley, M. (1993). Fractal everywhere. Second Edition. Academic Press, Inc., Boston.
[4] Fractal – Wikipedia.}

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NOTE: The figures 1 and 4 are self-created using Ultra Fractal 5.04 fractal for Windows, and Paint. The figure 3 was obtained by editing the original in Paint.

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I hope the post was of your interest and pleasure. If you have any question or suggestion, i invite you to leave your comment and i will gladly answer. Thanks for your kind reading!

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In future installments, I will be talking about historical and mathematical aspects of fractals, some of the classic fractals, about the types of fractals, among others.