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Characteristics Of Fractals And Fractal Dimensions Engineering Essay

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According to Benoit B. Mandelbrot, fractal is considered that object or structure that consists of fragments with variable orientation and size but of similar appearance. This feature gives the fractal some special geometric properties the length and the relationship between surface area and volume. These special properties do need other s different mathematical tools to explain the common characteristics. In the human body there are structures with fractal geometry, such as vascular system, the bronchial ramifications, the neural network, the arrangement of the glands, etc. The importance of this fractal geometry in the body is to optimize the role of systems because in a small space with the largest area. Since there are structures with fractal geometry we deduce that should be possible phenomena with fractal characteristics to power these phenomena have constantly repeating patterns at different timescales. These phenomena can be characterized with the use of mathematical tools of fractal geometry.

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.

The concept of dimension that we use is usually the classical Euclidean, is that one dimension is a line, form a flat two-dimensional and three-dimensional object form a volume.

However, an irregular line tends to form a surface and a surface bends when it becomes a volume, as we can, starting a one-dimensional object, passing the same object in three dimensions. Many natural structures have these characteristics so that, geometrically, these structures may have a non integer dimension between 2 and 3.

Thus the fractal dimension is an index that allows us to quantify the geometric properties of objects with fractal geometry. The phenomena with fractal behavior can be represented by line graphs, and these graphics can measure their fractal dimension and thus to quantify the complexity of chaotic dynamics.

Regarding the relationship between fractals and chaos, we could truly say that fractals are the graphic representation of chaos. Delving 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. The most curious part of this is that both the equations and fractals can be constructed with elements that we have all seen in our past academia, but the results obtained can become an incredibly high complexity. This can be considered a way of life...

Fractal Characteristics

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. However, in the seventies of last century, their study is closely linked to development studies on chaos.

As noted above, the fractals are basically the graphical representation of chaos, but also have a number of characteristics that then we will try to enumerate. 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 in nature. In fact, we will be surprised a lot when we stumble across, for example, with 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 originates 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 divide it into three parts and remove the core and so on.

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 book Gulliver's Travels when he conceived the idea of the existence of tiny men, the midgets, and giants, all with similar morphology but a quite different scale. Of course, this is very attractive and even romantic, but rejects the science for a long time. 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, is viewed as a fractal object every time we change the scale, 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 Koch curve, 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. In principle, we may consider Europe as a peninsula of Asia Moreover, 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 Mandelbrot How long is the coast of Britain? In which he proposes the concept of fractal dimension. According to Euclid's geometry, we move in a three-dimensional as to place a point on the plane we need three coordinates (height, width and depth). Similarly, a plane has two dimensions, the straight one and point zero. 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 the ruler 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, as shown in the following pages 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.

CALCULATION OF FRACTAL DIMENSIONS

As I mentioned above, we defined the concept of fractal dimension as one that does not fit, traditionally considered since the time of Euclid: size 0, item; dimension 1, the line, and so on. ... But this concept is not only theoretical but can be calculated as we will show below. Anyway, we should not forget that we start with a subjective idea, as it is to ascertain and quantify the fractal occupies the space where you are.

If we take a figure whose fractal dimension is between one and two as, for example, the coastline, the result of its length will depend on the length of the ruler we use, for example the unit of measurement. Therefore, if we get this unit to be infinitely small we can measure with great accuracy.Now, based on this simple idea, it will be easier to understand the following mathematical development:

Denote a complete metric space and (X, d), where is a nonempty compact subset of X. whereas take B (x,) as areas closed to radio and with center at a point xX.

We define an integer, N (A,) that is the least necessary number of areas closed to radio we need to cover all A.. This would be:

N (A,) = The smallest positive integer so that AÌ ÈMn=1 B(xn, e) For a set of distinct points (xn, 1, 2, 3, ..., M). To know that this number exists, surround all the points x A with an area open to radio > 0 to cover A with joint open. Since A is compact, this cover has a finite sub cover, which is an integer, which call M '. If we close these areas, we get a cover M 'of closed mats.

We call C the set of covers of A with a maximum of M 'areas closed to radio. Therefore, C contains at least one item. Now, let's f:C à {1, 2, 3,&,M } as f (c) which is equal to the number of areas on deck c C. Then, {f(c): cÎC} is a finite set of positive integers. Consequently, this set will contain a smaller number, N (A,).

Intuitive idea behind fractal dimension, based on the assumption that A has a fractal dimension D if N(A, e) » Ce -D where C is a positive constant. Interpret"» so that f ( ) and g () are real functions of real positive variable. Then, f(e) » g(e) Means that .

Solving for D we get:

Given that time tends to zero, we get the term also tends to zero we arrive at the following definition:

Be AÎH(X), and (X, d) is a metric space. For each e>0 let N (A, e) And lower number of area closed to radio?> 0 needed to cover A. If:

Exists, then D is the fractal dimension of A. Also denoted as D = D (A) and reads "A has fractal dimension D"

Examples:

We can recreate this set a very simple way: we take a line and divide it into three equal segments, eliminating the middle and replaced by two segments of a length equal to one third of the original line thus obtaining four segments, this is continued to infinity.

K

E

N

0

1

1

1

1 / 3

4

2

1 / 9

16

K

K = number of interactions required

E = size measuring instrument

N = Number of times used E

Its size is calculated using the following formula:

And which leads to:

Thus see that the dimension of the Koch curve has a dimension that is between the 1st and the 2nd and is 1.2618.

The main and most known representative of fractals is the Mandelbrot set. For many experts it is by far the most complex object of all sciences. 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. Of course it looks like a diabolical invention capable of driving the sanest.

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 and either C squared. We add the number obtained C and back to be squared and continue over and over again with the same process: z z2 + C.

Applications of Fractals

Although they may seem simple figures created to entertain mathematicians, there are many applications of fractals, both theoretically and practically. Given the broad scope of its application field, then we will limit to list the most striking and, so to speak, which are more spectacular.

Since then, its application in the field of abstract science has been really great. One of its most immediate applications is the study of solutions of systems of equations over the second degree. In fact, early in the study of fractals, John Hubbard, American mathematician, in a plane represent the way the Newton method for solving equations, leads from different starting points for each of the solutions. Previously it was thought that each solution will have a basin of attraction that would divide the map in several places and points of which lead to the solution. However, by computer scanning and assigning a color to each watershed, Hubbard found that the boundaries of these regions of the plane were not well defined in any way. Within these limits was a color points into other points of color and as the grid of numbers was more complex was going to expand revealing the border. In fact, could be considered as there was no such border.

Although there are many applications in areas as diverse as physics and seismology, since then the area where more applications have been found in image processing. In fact, rather than inputs, should speak of a revolution. Michael Barnsley was 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 funds 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.


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