Fractal art
Fractal by Lynn (Gracie's mom)
Fractal by Lynn (Gracie's mom)
Fractal
From Wikipedia, the free encyclopedia
In nature also appears fractal geometry, as in this romanescu.
El término fue propuesto por el matemático Benoît Mandelbrot en 1975 y deriva del Latín fractus , que significa quebrado o fracturado. A fractal is a semi-geometric object whose basic structure, fragmented or irregular, is repeated at different scales. [1] The term was proposed by the mathematician Benoît Mandelbrot in 1975 and Fractus derives from the Latin, meaning broken or fractured. Many natural structures are kind of fractal.
A fractal is a geometric object attributed the following characteristics [2]
- It is too irregular to be described in traditional geometric terms.
- It has detail on any scale of observation.
- It was autosimilar (exact or approximate statistically).
- Their Hausdorff-Besicovitch dimension is strictly larger than its topological dimension.
- Is defined by a simple recursive algorithm.
We are not just one of these characteristics to define a fractal. For example, the line is not considered a real fractal, because despite being an object autosimilar lacks the rest of characteristics required.
A fractal is a natural element of nature that can be described using fractal geometry. o los copos de nieve son fractales naturales. The clouds, mountains, the circulatory system, coastlines [3] or snowflakes are natural fractals. This representation is approximate, because the properties attributed to objects fractals ideals, such as infinite detail, have limits on the natural world.
Table of Contents |
Introduction [edit]
The classic examples [edit]
To find the first examples of fractals must go back to the late nineteenth century: in 1872 came the role of Weierstrass, whose graph today fractal consider, as an example of continuous but not differentiable function at any point.
Successive steps in the construction of the Koch curve
Later examples appeared with similar properties but a more geometric. Such examples could be constructed starting from an initial figure (seed), which applied a series of simple geometric constructions. The series of figures obtained approached a figure that corresponded to the limit we now call fractal set. Thus, in 1904, Helge von Koch defined a curve with properties similar to that of Weierstrass: the Koch snowflake. In 1915, Waclaw Sierpinski built his triangle, and a year later, his carpet.
| Construction of the Sierpinski carpet: | |||||
 |  |  |  |  | |
| Step 1 (seed) | Step 2 | Step 3 | Step 4 | Step 5 | |
These sets showed the limitations of classic analysis, but were seen as artificial objects, a "gallery of monsters," as called Poincaré. Few mathematicians saw the need to study these objects in themselves. [4]
In 1919 there is a basic tool in the description and extent of these assemblies: the Besicovitch-Hausdorff dimension.
Julia sets [edit]
These sets, the result of the work of Pierre Fatou and Gaston Julia in the 1920s, arising as a result of repeated application of functions holomorfas 
.
Consider the case of polynomial functions of greater than one. By applying successive times a polynomial function is very possible that the result tends to 
. The set of values 
not beyond the infinite through this operation is called Julia set of stuffing, and their border, simply set of Julia.
These sets are represented by an algorithm of time window in which each pixel is colored by the number of iterations required to escape. Often used a particular color, often black, to represent the points that have not escaped after a large number of iterations and default.
Examples of Julia sets for f c (z) = z 2 + c
 Julia set of stuffing associated af c, c = (φ-2) + (φ-1) i =- 0.4 +0.6 i |  Julia set of stuffing associated af c, c =- 0.835-0.2321i |
In black, image of the Mandelbrot set overlapped with the sets of Julia stuffed represented by some of its points (on the sets of red and blue Julia related non-related).
Families of fractals: the set of Mandelbrot [edit]
presenta conjuntos de una variedad sorprendente. The family of Julia sets of f (c), associated with the recurrence of functions of the form f c (z) = z 2 + c presents a surprising variety of packages.
That family will have special significance to be parameterized on a map of fractal called the Mandelbrot set. This set M represents a map in which each pixel corresponding to a value of the parameter 
, Is colored so that it reflects a basic property of all Julia associated with f c. Specifically, 
if the Julia set of f c is associated with related.
Characteristics of a fractal [edit]
Autosimilitud [edit]
Autosimilitud exact snowflake Koch
According to B. Mandelbrot, an object was autosimilar or autosemejante if their parties have the same shape or structure of the whole, though they may stand for different scale and may be slightly deformed. [5]
Fractals may have three types of autosimilitud:
- Autosimilitud accurate. This is the most restrictive autosimilitud: demands that the fractal look identical on different scales. Often found in the fractal-defined system functions ITER (IFS).
Cuasiautosimilitud in the Mandelbrot set: to vary the scale we get copies of the set with minor differences.
- Cuasiautosimilitud: demands that the fractal seems about the same at different scales. Fractals contain such minors and distorted copies of themselves. D. Sullivan mathematically defined the concept of joint-cuasiauto similar to the concept of quasi-isometry. Fractals defined by relations of recurrence are usually of this type.
- Autosimilitud statistics. It is the weakest type of autosimilitud: it demands that the fractal has numerical or statistical measures that are preserved with the change of scale. Fractals are random examples of fractals of this type.
Dimension fractal dimension and Hausdorff-Besicovitch [edit]
Among the fractal curve as we can find examples that fill the entire plane. In that case, the topological dimension of the curve, which is a not inform us about how this is the space environment. In general, we might ask how a set of densely occupies the metric space that contains it. The numbers tell us objectively of such issues are:
- The fractal dimension. The formulas that define it have to do with the counting of the balls needed to cover all or with a grid of boxes containing part of the package when the size of some tend to zero. We can measure the fractal dimension of real objects: coastlines (1.2), clouds, trees, etc. With these measures we can compare real-world objects with fractal generated by mathematical algorithms.
- The dimension of Hausdorff-Besicovitch. It has a more complex than the fractal dimension. Its definition is not usually used to compare sets of the real world.
Autosimilitud statistics of a fractal generated by the process of diffusion limited aggregation
Defined by recursive algorithms [edit]
We can highlight three common techniques to generate fractals:
- Systems functions ITER (IFS). Some sets are replaced by recursively its image under a system of applications: the set of Cantor, the Sierpinski carpet, the Sierpinski triangle, the Peano curve, the curve of the dragon, the Koch snowflake or the Menger Sponge , Are examples.
- Fractals time window, defined by a ratio of recurrence at each point in space (for example, the complex): the Mandelbrot set, a set of Julia, and the fractal Lyapunov.
- Fractal random processes generated by stochastic, not deterministic: the Brownian movement, the Lévy flight, landscapes fractal Brownian or trees. The latter are produced by processes of diffusion limited aggregation ..
Mathematical aspects [edit]
Attempts to rigorous definition [edit]
The concept of fractal not available in 2008 for a precise mathematical definition and acceptance. Partial attempts to give a definition were made by:
- B. Mandelbrot, which in 1982 defined as a set whose fractal dimension of Hausdorff-Besicovitch is strictly larger than its topological dimension. He himself acknowledged that his definition is not sufficiently large.
- D. Sullivan, who mathematically defined a category of fractals with its definition of joint cuasiautosimilar that made use of the concept of quasi-isometry.
Fractal dimension [edit]
necesarias para recubrir el conjunto, como el límite: Can be defined in terms of the minimum number N (ε) ball of radius ε needed to cover the set as the limit:
que intersecan al conjunto: Or depending on the count of the number of boxes N n of a grid of width 1 / 2 n that intersect the whole:
We show that both definitions are equivalent, and that are invariant under isometrics. [6]
Dimension Hausdorff-Besicovitch [edit]
es la siguiente: In a more complex dimension of Hausdorff-Besicovitch gives us a number D H (A), also invariant under isometrics, whose relationship with the fractal dimension D F (A) is as follows:
This allows in some cases to distinguish between sets with the same fractal dimension.
Fractal dimension resulting from an IFS [edit]
In that case, where there is overlap, it is shown that D F D = M and both can be calculated as a solution of the equation:
where c i desgna factor contraction of each application of the contractionary IFS.
Applications [edit]
We used fractal techniques in data compression and in various scientific disciplines.
Image compression [edit]
Compress the image of an object such as the fern autosemejante figure is not difficult: making use of collage theorem, we must find an IFS, a set of transformations which carries the full figure (in black) in each of its parts autosemejantes (red , Azure and dark blue). The information on the image is encoded in the IFS, and repeated application of these transformations to obtain the processed image in question.
But the previous approach poses real problems with many images: Do not expect that, for example, the image of a cat kittens This little distorted on himself. To solve, Arnaud Jacquin in 1989 created the outline of systems iterator functions partition: it is divided into the image through a partition and for each region resulting seeks another region similar to the first under the appropriate tranformaciones. [7]
The resulting pattern is a compression system at a loss, asymmetrical time. Unfortunately still takes a long time to find the transformations that define the image. However, once found, the decoding is very fast. Compression, though influenced by many factors, often comparable to JPEG compression, so the time factor is crucial to opt for either system.
Modeling natural forms [edit]
Fraction of a fractal Mandelbrot
The forms fractals, the ways in which the parties are close to everything, are present in the biological material, along with the symmetries (basic forms that require only half of genetic information) and spirals (The forms of growth and development the basic shape to the occupation of a larger space), as the most sophisticated in the evolutionary development of the biological material in terms that are presented in processes in which they are produced qualitative leaps in the biological ways, that is possible disasters ( extraordinary events) that give rise to new realities more complex, such as leaves that have a morphology similar to the small branch that are part of that, in turn, have a way similar to the branch, which is similar to the shape of the tree, but qualitatively it is not the same one sheet (biological simple form), a branch or a tree (how complex biological).
Dynamical systems [edit]
A strange attractor: the Lorenz Attractor
But also forms fractals are not only in the forms of space objects, but what you see on the evolutionary dynamics of complex systems (see chaos theory). Dynamic that consists of cycles (which from a simple reality set in just creating a new reality more complex) which in turn are part of more complex cycles that in turn form part of the development of the dynamics of another major cycle, which .... and the dynamic developments of all these cycles show the similarities of the chaotic systems themselves.
In artistic events [edit]
Literature and poetry [edit]
It was also used as a union between art and science, this is an example of the scientist-German-Chilean poet Mario Markus.
Graphic arts [edit]
With software like Apophysis or Ultra Fractal images can be made with various techniques, changing parameters, geometry of triangles or random changes (sometimes called "mutations").
Extrapolation to concepts of Social Sciences [edit]
Several individuals can now take advantage of science concepts of the theory of fractals in their respective areas of expertise. An example is found in the social sciences. [8]
Extrapolate too schematic of fractal geometry to the social sciences will always be a utopia, because society is not just a mathematical abstraction. A society can not find an equation that generates a summary given by the simple fact that the pillars of society is more elastic than just coordinates ideals. Yes that is what the chaos theory is called "extreme sensitivity" to the "initial state" of a process, which can result in drastic changes after a while the start, as postulated the theory of Chaos, Can not a crisis economic (national) impact on the entire system of the world economy?
With the study of the human genome, which is trying to do is get the laws governing the development of human beings, making it possible to predict phenomena that were previously impossible to study. However, society does not have a "DNA" so rigid as human beings. The analysis of the "social DNA", or all of its domestic development trends, can be done by following the parameters of this theory. Put another way, is a novel way that you can take a dialectical way that Marx founded.
Marx also studied other equations summary that engendered the global capitalist structure: One was the private ownership of the means of production. She studied how to develop this historical phenomenon. And drew the conclusion that private ownership tended to monopoly. But was unable to determine "exactly" the future of the system, and that capitalism does not have a DNA allowing to accurately predict its enduring development, historic. And if I had, in times of Marx nobody understood yet.
Therefore, the social sciences are among the beat hard and soft sciences. It was not a "hard science" by the inability to find precise laws. But laws can find elastic, bringing the object of study without sacrificing the science. The method can be used for this is the chaos theory and fractals.
In this relate the theory of fractals and chaos theory, which are part of a new and emerging paradigm in Science. The theory of systems of Ludwig von Bertalanffy also has to make their contributions, like the theory of catastrophes, Rene Thom.
