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Niels Fabian Helge von Koch said, Fractal theory can be considered a valid and useful tool for studying dynamic phenomena in the human body or in nature and allows an approach more in keeping with the complexity and nonlinearity existing in these processes. The fractal dimension is a mathematical index that we calculate and that allows us to quantify the characteristics of fractal objects or phenomena. This index can be calculated in several ways. One of these ways of calculating fractal dimension is the Hurst exponent, which is a number that indicates the degree of influence of the present over the future (degree of similarity of the phenomenon with the "Brownian movement".)
The concept of dimension that we use is usually the classical Euclidean. Fractal dimension is an index that allows us to quantify the geometric properties of objects with fractal geometry. The fractal behavior can be represented by line graphs, and these graphs can measure their fractal dimension and thus to quantify the complexity of chaotic dynamics.
The relationship between fractals and chaos, we could truly say that fractals are the graphic representation of chaos. Exploring a bit on the subject and based on the ideas of Carlos Sabino we could say that the relationship between chaos and fractals is that fractals are geometric figures with a certain pattern that is repeated endlessly as a multiple scales and if the close look reveals that this pattern is found in the components, and parts of its components, and component parts of its components, and so on to infinity. This we can see if we can observe the fractal at different scales smaller and smaller.
Fractals of which is said not to have full dimension represent graphically that chaotic equations can be solved. Fractals show us that points of a given mathematical space collapsed the chaotic solutions of our equation.
In broad terms we can define a fractal as a geometric figure with a very complex and detailed structure at all scales. Already in the nineteenth century many figures were designed with these characteristics but were not considered beyond simple mathematical curiosities and rarities.
Fractals are basically the graphical representation of chaos, but also have a number of characteristics. First, we must consider that they are still fractal geometric figures, but do not meet its definition and it is impossible through traditional concepts and methods in place since Euclid. However, the above statement is very far from becoming rare or anomalous figures, as a glance around us can perceive the lack of Euclidean forms ideal, a feeling which will increase greatly if we find fractals in nature. In fact, we will be surprised a lot when we stumble across, for example, a spherical stone. Consequently, while always trying to apply to reality, Euclidean shapes (circles, squares, cubes ...) are limited to the field of our mind and the pure mathematical abstraction. On the contrary, as we shall see, fractals are widespread.
Like when we speak of chaos, one of the most significant properties of fractals and which is particularly striking is the fact that fractals originate from some initial conditions or very basic rules that will lead to extremely complex shapes, seemingly diabolical. A clear example is the Cantor set, because it originates simply part of a line segment, we remove the center core and repite this process on the remaining pieces as illustrated below.
Another key feature of the concept of fractal self-similarity is. This idea in a broader sense and philosophy has attracted since the beginning of man's humanity. Jonathan Swift partly reflected in his bookÂ Gulliver's TravelsÂ when he conceived the idea of the existence of tiny men, theÂ Lilliputians, and giants, all with similar morphology but a quite different scale. However, the advances of this century that unveiled some resemblance of an atom with electrons orbiting around the nucleus and the solar system with the Sun and its planets rehabilitated to some extent the concept. In the particular case of fractals, it is viewed as a fractal object every time we change the scale of an object, and this shows a clear resemblance to the previous image. Therefore, we can define the self-similarity as symmetry within a scale, in other words fractals are recurrent.
This is evident in figures like the Sierpinski triangle, in which each extension results in an exact copy of the picture above. But to illustrate in a general way, we can see the coastline of Europe. Within Europe there are large peninsulas and the Balkans and if we reduce the scale, we found other small and the Peloponnese peninsula and we can continue to differentiate between incoming and outgoing calls between the grains sand from the beach.
However, this self-similarity should not be confused with an absolute identity between scales, for example, following the previous example, is not that smaller peninsulas have a way exactly like the majors. Rather, what this idea implies the existence of an infinite complexity of fractal figures since, given its recurrence, we will be extending its image over and over again to infinity without the appearance of a completely defined. In fact, these extensions will be revealing an increasingly complex network and seemingly inexplicable. For example, we take a seemingly smooth surface but if we extend it, the microscope will show hillocks and valleys that will be more abrupt increases as we use more.
But this discovery leads us to a more difficult question, what is the size of a fractal? This same question was asked in his article MandelbrotÂ How long is the coast of Britain?Â In which he proposes the concept of fractal dimension. According to Euclid's geometry, we move in a three-dimensional place; to plot a point on the plane we need three coordinates (height, width and depth). Similarly, a plane has two dimensions. However, if we take, for example, the Koch curve is assumed to belong to a one-dimensional world, we will see as their length varies depending on theÂ rulerÂ that we use and, therefore, it is impossible to calculate exactly. Clearly, neither is it a plane because as its name suggests is a curve as it is within the plane. Consequently, it is considered that its size must be halfway between one and two.
This approach may seem a simple mathematical juggling, since this unit the size of the unit of measure and, ultimately, of the relativity of the reference point of the observer escapes hands. However, it is very useful because it can be calculated and, therefore, serves to balance characteristics of fractal objects and their degree of ruggedness, discontinuity or irregularity. This also means that it is considered that this degree of irregularity is constant at different scales, which has been shown many times appearing incredibly regular and irregular patterns of behavior in the complete disorder.
Examples in nature:
All throughout this world we live surrounded by fractals. A perfect example of an always moving fractal is the human being. Other examples of fractals are natural occurring events like lighting, snow, the shape of growing plants and many others.
The main and most well-known representative of fractals is the Mandelbrot set. For many experts it is by far the most complex object of all. It is amazing to observe its infinite complexity, which is certainly beyond description. And this complexity is multiplied at every scale clusters appear endless, peninsulas, islands really are not, spirals, etc. No matter how scaling up or how many times you give to the zoom button, the display will appear more and more figures infinitely complicated.
The Mandelbrot set is a series of complex numbers that satisfy a certain mathematical property. Each issue is composed of a real and an imaginary part represented by i, which is equal to the square root of -1, as follows: 2 + 3i. So take a number C, then squared. Then we add the number obtained C, back to the squared and continue over and over again with the same process:
Applications of Fractals
Although they may seem simple figures created to entertain mathematicians, there are many applications of fractals, both theoretically and practically. One of its most immediate applications is the study of solutions of systems of equations over the second degree.
Other practical uses of fractals are in the computer science field like for instance fractal archivation and image processing. Michael Barnsley is the pioneer in the treatment of images from its so-called fractal transformation. This is the opposite process to the formation of a fractal, for example instead of creating a figure from certain rules; we search for rules that form a specific figure.
Currently, fractals are used to compress digital images so that they occupy less space and can be transmitted at higher speed and lower cost; in addition, they are very useful when creating spectacular special effects blockbusters, because it is relatively easy to create all types of landscapes and backgrounds through fractals. So simple that with a small computer program that occupies a small space, you can create a beautiful tree from a simple scheme.
Similarly, the fractal revolution affects the world of music, as it is very widespread use of fractal procedures for the composition, especially techno music or rhythmic foundation for any other type of music.
Furthermore, the concept of fractal dimension and have had great impact in the field of biology. On the other hand, one can see great examples of fractal structures in the human body as the network of veins and arteries. From a large blood vessel and the aorta come out smaller vessels until the appearance of very fine hair so as to cover as much space as possible to carry nutrients to cells. Furthermore, it is believed to guess a certain similarity between the generation of fractals and the genetic code, since in both cases from very limited information apparently complex structures arise.