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Characteristics Of Fractals And Fractal Dimensions Engineering Essay

According to Benoit B. Mandelbrot, fractal is considered that thing or composition that contains fragments with changing 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 size. These special properties do need other s different mathematical tools to make clear the normal characteristics. In our body there are set ups with fractal geometry, such as vascular system, the bronchial ramifications, the neural network, the arrangement of the glands, etc. The need for this fractal geometry in the body is to improve the role of systems because in a small space with the greatest area. Since there are buildings with fractal geometry we deduce that needs to be possible phenomena with fractal characteristics to force these phenomena have constantly duplicating habits at different timescales. These phenomena can be characterized by using numerical tools of fractal geometry.

Niels Fabian Helge von Koch said, "Fractal theory can be viewed as a valid and useful tool for studying strong phenomena in our body or in nature and allows a strategy 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 things or phenomena. This index can be determined in a number of ways. One of these ways of calculating fractal aspect is the Hurst exponent.

The idea of dimension that we use is usually the classical Euclidean, is the fact one aspect is a range, form a set two-dimensional and three-dimensional thing form a volume.

However, an irregular line tends to form a surface and a surface bends when it becomes a level, once we can, starting a one-dimensional thing, moving the same subject in three measurements. Many natural constructions have these characteristics so that, geometrically, these buildings may have a non integer dimensions between 2 and 3.

Thus the fractal dimensions is an index that allows us to quantify the geometric properties of things with fractal geometry. The phenomena with fractal behavior can be represented by range graphs, and these design can measure their fractal dimension and thus to quantify the complexness of chaotic dynamics.

Regarding the relationship between fractals and chaos, we could truly say that fractals will be the visual representation of chaos. Delving a bit about them and predicated on the ideas of Carlos Sabino we're able to say that the relationship between chaos and fractals is the fact that fractals are geometric information with a certain structure that is repeated endlessly as a multiple scales in case the close look reveals that pattern is situated in the components, and parts of its components, and component elements of its components, etc to infinity. This we can easily see if we can observe the fractal at different scales smaller and smaller.

Fractals of which is said not to have full sizing stand for graphically that chaotic equations can be resolved. Fractals show us that points of a given mathematical space collapsed the chaotic solutions of our equation. The most interested part of this is that both equations and fractals can be designed with elements that we have all seen in our past academia, but the results obtained may become a remarkably high complexity. This can be considered a means of life. . .

Fractal Characteristics

In broad terms we can identify a fractal as a geometric physique with a very complex and specific structure at all scales. Already in the nineteenth hundred years many results were made 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 known above, the fractals are simply the graphical representation of chaos, but also have a number of characteristics that then we will attempt to enumerate. First, we should consider they are still fractal geometric results, but do not meet its description and it is impossible through traditional concepts and methods in place since Euclid. However, these statement is very far from becoming exceptional or anomalous numbers, as a look all around us can perceive the lack of Euclidean forms ideal, a sense that may increase greatly if we find in character. In fact, we will be surprised a great deal when we stumble across, for example, with a spherical natural stone. Therefore, while always trying to use to actuality, Euclidean figures (circles, squares, cubes. . . ) are limited to the field of our mind and the clean mathematical abstraction. On the other hand, as we will see, fractals are widespread.

Like whenever we talk about chaos, one of many properties of fractals and which is specially striking is the fact that hails from some first conditions or very basic guidelines that will lead to extremely sophisticated shapes, relatively diabolical. A definite example is the Cantor collection, since it originates simply part of the line segment, we separate it into three parts and remove the core and so forth.

Another key feature of the concept of fractal self-similarity is. . . This notion in a broader sense and school of thought has attracted since the start of man's mankind. Jonathan Swift partly mirrored in his booklet Gulliver's Travels when he conceived the idea of the presence of small men, the midgets, and giants, all with similar morphology but a quite different range. Of course, this is very attractive and even passionate, but rejects the technology for a long period. However, the advances of this hundred years that launched some resemblance of atom with electrons orbiting surrounding the nucleus and the solar system with the Sun and its planets rehabilitated to some extent the concept. In this case of fractals, can be regarded as a fractal object whenever we change the size, shows a definite resemblance to the previous image. Therefore, we can explain the self-similarity as symmetry within the scale, in other words fractals are repeated.

This is noticeable in figures like the Koch curve, in which each extension results within an exact copy of the picture above. But to demonstrate in a general way, we can see the coastline of European countries. In principle, we might consider European countries as a peninsula of Asia Furthermore, within Europe there are large peninsulas and the Balkans and if we decrease the scale, we found other small and the Peloponnese peninsula and we can continue steadily to differentiate between incoming and outgoing cell phone calls between your grains sand from the beach.

However, this self-similarity shouldn't be confused with a complete individuality between scales, for example, following previous example, isn't that smaller peninsulas have a means exactly like the majors. Alternatively, what this notion implies the existence of an infinite complexity of fractal results since, given its recurrence, we will be increasing its image over and over again to infinity without the looks of a completely defined. In fact, these extensions will be uncovering an increasingly intricate network and seemingly inexplicable. For example, we take a seemingly clean surface but if we lengthen it, the microscope will show hillocks and valleys that'll be more abrupt rises once we use more.

But this breakthrough leads us to a more difficult question, what's how big is a fractal? This same question was asked in his article Mandelbrot The length of time is the coast of Britain? In which he proposes the idea of fractal dimension. Regarding to Euclid's geometry, we move in a three-dimensional as to place a point on the airplane we are in need of three coordinates (height, width and depth). Likewise, a planes has two measurements, the right one and point no. However, if we take, for example, the Koch curve is assumed to belong to a one-dimensional world, we will see as their size differs depending on the ruler that people use and, therefore, it is impossible to estimate exactly. Clearly, nor is it a plane because as its name implies is a curve as it is at the plane. Therefore, it is considered that its size must be halfway between one and two.

This strategy may seem a straightforward numerical juggling, since this unit how big is the machine of solution and, finally, of the relativity of the guide point of the observer escapes hands. However, it's very useful because, as shown in the next pages can be computed and, therefore, acts to balance characteristics of fractal items and their degree of ruggedness, discontinuity or irregularity. This also means that it is considered that amount of irregularity is constant at different scales, which includes been shown often appearing amazingly regular and unusual patterns of patterns in the entire disorder.

CALCULATION OF FRACTAL DIMENSIONS

As I mentioned previously, we defined the idea of fractal dimensions as one that does not fit, typically considered because the time of Euclid: size 0, item; aspect 1, the series, and so forth. . . . But this idea isn't just theoretical but can be determined even as we will show below. In any case, we should remember that we focus on a subjective idea, as it is to see and quantify the fractal occupies the space what your location is.

If we have a shape whose fractal aspect is between one and two as, for example, the coastline, the consequence of its span will rely upon the space of the ruler we use, including the unit of dimension. Therefore, if we get this product to be infinitely small we can assess with great exactness. Now, based on this simple idea, it'll be better to understand the following mathematical development:

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

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

N (A, ) = The smallest positive integer so that A Mn=1 B(xn, e) For a couple of distinct details (xn, 1, 2, 3, . . . , M). To learn that this number exists, encompass all the tips x A with an area available to radio > 0 to cover A with joint open up. Since A is small, this cover has a finite sub cover, which can be an integer, which call M '. If we close these areas, we get a cover M 'of finished mats.

We call C the group of covers of your with a maximum of M 'areas finished to radio. Therefore, C consists of at least one item. Now, let's f:C   1, 2, 3, &, M as f (c) which is equal to the amount of areas on deck c C. Then, f(c): cC is a finite group of positive integers. As a result, this collection will contain a smaller number, N (A, ).

Intuitive idea behind fractal aspect, predicated on the assumption that A has a fractal aspect D if N(A, e) » Ce -D where C is a confident 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 explanation:

Be AH(X), and (X, d) is a metric space. For each e>0 let N (A, e) And lower amount of area closed down to radio?> 0 had a need to cover A. If:

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

Examples:

We can recreate this collection a simple way: we take a line and separate it into three equivalent segments, eliminating the center and substituted by two segments of a period equal to 1 / 3 of the original line thus obtaining four sections, this is extended to infinity.

K

E

N

0

1

1

1

1 / 3

4

2

1 / 9

16

K

K = number of interactions required

E = size calculating instrument

N = Amount of that time period used E

Its size is calculated using the following formula:

And that leads to:

Thus see that the dimensions of the Koch curve has a aspect that is between the 1st and the 2nd which is 1. 2618.

The main & most known agent of fractals is the Mandelbrot set. For most experts it is by far the most complex object of all sciences. It is amazing to observe its infinite complexity, which is obviously beyond description. And this complexity is multiplied at every size clusters appear limitless, peninsulas, islands really are not, spirals, etc. Regardless of how scaling up or just how many times you share with the move button, the screen will appear increasingly more numbers infinitely complicated. Certainly it appears like a diabolical technology capable of traveling the sanest.

The Mandelbrot collection is some complex amounts that fulfill a certain mathematical property. Each issue comprises a genuine and an imaginary part symbolized by i, which is add up to the square root of -1, as follows: 2 + 3i. So have a quantity and either C squared. We add the quantity obtained C and back again to be squared and continue over and over again with the same process: z z2 + C.

Applications of Fractals

Although they may seem simple numbers created to captivate mathematicians, there a wide range of applications of fractals, both theoretically and pretty much. Given the wide-ranging opportunity of its application field, then we will limit to list the most impressive and, as they say, which are more stunning.

Since then, its software in the field of abstract science has been excellent. One of its most immediate applications is the study of solutions of systems of equations over the next degree. In fact, early in the study of fractals, John Hubbard, North american mathematician, in a airplane represent what sort of Newton way for dealing with equations, leads from different starting items for every of the solutions. Recently it was thought that each solution will have a basin of fascination that would separate the map in several places and points of which lead to the answer. However, by computer scanning and assigning a color to each watershed, Hubbard found that the boundaries of these parts of the plane were not well defined in any way. Within these boundaries was a color tips into other tips of color and as the grid of volumes was more complex was going to expand revealing the border. In fact, could be considered as there was no such boundary.

Although there are many applications in areas as diverse as physics and seismology, since then the region where more applications have been within image processing. Actually, alternatively than inputs, should speak of a trend. Michael Barnsley was the pioneer in the treating images from its so-called fractal transformation. This is actually the contrary process to the formation of a fractal, for example instead of creating a body from certain rules; we seek out rules that form a particular figure.

Currently, fractals are used to compress digital images so that they occupy less space and can be transmitted at higher velocity and lower cost; in addition, they are incredibly useful when creating spectacular special results blockbusters, since it is not too difficult to create all types of landscapes and cash through fractals. So simple that with a small computer program that occupies a tiny space, you can create a lovely tree from a simple scheme.

Similarly, the fractal revolution influences the world of music, as it's very wide-spread use of fractal techniques for the composition, especially techno music or rhythmic base for any other type of music.

Furthermore, the concept of fractal dimension and also have got great impact in neuro-scientific biology. On the other hand, one can see great examples of fractal structures in the body as the network of veins and arteries. From a sizable blood vessel and the aorta turn out smaller vessels before appearance of very fine mane to be able to cover all the space as is feasible to carry nutrition to cells. Furthermore, it is believed to guess a certain similarity between your era of fractals and the genetic code, since in both cases from very limited information apparently complicated structures arise.

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