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FRACTALSIn Nature, Arts andScience Dr. Farhana Shaheen Assistant Professor YUC- Women Campus
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FRACTALS???For centuries, mathematicians rejectedcomplex figures, leaving them under asingle description: ―formless‖. Forcenturies, geometry was unable todescribe trees, landscapes, clouds, andcoastlines. However, in the late 1970’s arevolution of our perception of the worldwas brought by the work of BenoitMandelbrot who introduced FRACTALS.
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―Fractua‖ means IrregularFractals are geometric figures likecircles, squares, triangles etc., buthaving special properties. They areusually associated with irregulargeometric objects, that look the sameno matter at what scale they areviewed at.A fractal is an object in which theindividual parts are similar to thewhole.
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Fractals exhibit self-similarityFractals have the property of self-similarity, generated by iterations,which means that various copies ofan object can be found in the originalobject at smaller size scales.The detail continues for manymagnifications -- like an endlessnesting of Russian dolls within dolls.
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What is a Fractal? A fractal is a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole. The core ideas behind it are of feedback and iteration. The creation of most fractals involves applying some simple rule to a set of geometric shapes or numbers and then repeating the process on the result. This feedback loop can result in very unexpected results, given the simplicity of the rules followed for each iteration. Fractals have finite area but infinite perimeter.
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Examples of Fractals A cauliflower is a perfect example of a fractal where each element is a perfect recreation of the whole.
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A naturally occurring CauliflowerFractal Take a close look at a cauliflower: Take a closer look at a single floret (break one off near the base of your cauliflower). It is a mini cauliflower with its own little florets all arranged in spirals around a centre.
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Computer-generatedFractal patterns These days computer-generated fractal patterns are everywhere. From squiggly designs on computer art posters to illustrations in the most serious of physics journals, interest continues to grow among scientists and, rather surprisingly, artists and designers.
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The Sierpinski Triangle Lets make a famous fractal called the Sierpinski Triangle. Step One Draw an equilateral triangle with sides of 2 triangle lengths each. Connect the midpoints of each side. How many equilateral triangles do you now have?
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Cut out the triangle in thecenter.Step TwoDraw another equilateral triangle with sidesof 4 triangle lengths each. Connect themidpoints of the sides and cut out thetriangle in the center as before.
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The Sierpinski Triangle Unlike the Koch Snowflake, which is generated with infinite additions, the Sierpinski triangle is created by infinite removals. Each triangle is divided into four smaller, upside down triangles. The center of the four triangles is removed. As this process is iterated an infinite number of times, the total area of the set tends to infinity as the size of each new triangle goes to zero.
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Theory of Fractals Mandelbrot introduced and developed the theory of fractals -- figures that were truly able to describe these shapes. The theory was continued to be used in a variety of applications. Fractals’ importance is in areas ranging from special TV effects to economy and biology.
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Mandelbrot’sdiscovery The term fractal was coined by Benoit Mandelbrot in 1975 in his book Fractals: Form, Chance, and Dimension. In 1979, while studying the Julia set, Mandelbrot discovered what is now called the Mandelbrot set and inspired a generation of mathematicians and computer programmers in the study of fractals and fractal geometry.
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The Mandelbrot Set Named after Benoit Mandelbrot, The Mandelbrot set is one of the most famous fractals in existence. It was born when Mandelbrot was playing with the simple quadratic equation z=z2+c. In this equation, both z and c are complex numbers. In other words, the Mandelbrot set is the set of all complex c such that iteration z=z2+c does not diverge.
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The Julia set The Julia set is another very famous fractal, which happens to be very closely related to the Mandelbrot set. It was named after Gaston Julia, who studied the iteration of polynomials and rational functions during the early twentieth century, making the Julia set much older than the Mandelbrot set.
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Difference between the Julia setand the Mandelbrot set The main difference between the Julia set and the Mandelbrot set is the way in which the function is iterated. The Mandelbrot set iterates z=z2+c with z always starting at 0 and varying the c value. The Julia set iterates z=z2+c for a fixed c value and varying z values. In other words, the Mandelbrot set is in the parameter space, or the c-plane, while the Julia set is in the dynamical space, or the z-plane.
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Lorenz Model The Lorenz Model, named after E. N. Lorenz in 1963, is a model for the convection of thermal energy. This model was the very first example of another important point in chaos and fractals, dissipative dynamical systems, otherwise know as strange attractors.
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Objects in NatureMany objects in nature aren’t formed ofsquares or triangles, but of morecomplicated geometric figures. e.g.trees, ferns, clouds, mountains etc. areshaped like fractals. Other examplesinclude snowflakes, crystals, lightning, rivernetworks, cauliflower or broccoli, andsystems of blood vessels and pulmonaryvessels. Coastlines may also beconsidered as fractals in nature.
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Similarity between fractals andobjects in nature. One of the largest relationships with real- life is the similarity between fractals and objects in nature. The resemblance of many fractals and their natural counter- parts is so large that it cannot be overlooked. Mathematical formulas are used to model self similar natural forms. The pattern is repeated at a large scale and patterns evolve to mimic large scale real world objects.
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Fractals in Nature As fractals are patterns that reveal greater complexity as it is enlarged, they portray the notion of worlds within worlds. Trees and ferns are fractals in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees.
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Natural fractal pattern - airdisplacing a vacuum formed bypulling two glue-covered acrylicsheets apart.
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Fractal Geometry Fractal geometry is a new language used to describe, model and analyze complex forms found in nature. Chaos science uses this fractal geometry. Fractal geometry and chaos theory are providing us with a new way to describe the world.
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Fractal Geometry While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal geometry deals with objects in non- integer dimensions. Euclidean geometry is a description for lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithms -- a set of instructions on how to create a fractal.
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Applications of fractals in scienceFractals have a variety of applications in sciencebecause its property of self similarity existseverywhere. They can be used to model plants,blood vessels, nerves, explosions, clouds,mountains, turbulence, etc. Fractal geometrymodels natural objects more closely than doesother geometries.Engineers have begun designing and constructingfractals in order to solve practical engineeringproblems. Fractals are also used in computergraphics and even in composing music.
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Application of fractals and chaos isin music Some music, including that of Back and Mozart, can be stripped down so that is contains as little as 1/64th of its notes and still retain the essence of the composer. Many new software applications are and have been developed which contain chaotic filters, similar to those which change the speed, or the pitch of music.
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Special features of fractals A fractal often has the following features: It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. It is self-similar (at least approximately or stochastically). It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). It has a simple and recursive definition.
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Application to biological analysis Fractal geometry also has an application to biological analysis. Fractal and chaos phenomena specific to non-linear systems are widely observed in biological systems. A study has established an analytical method based on fractals and chaos theory for two patterns: the dendrite pattern of cells during development in the cerebellum and the firing pattern of intercellular potential. Variation in the development of the dendrite stage was evaluated with a fractal dimension. The order in many ion channels generating the firing pattern was also evaluated with a fractal dimension, enabling the high order seen there to be quantized.
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Real-Life Relevance AndImportance of Fractals and FractalGeometry Fractals have and are being used in many different ways. Both artist and scientist are intrigued by the many values of fractals. Fractals are being used in applications ranging from image compression to finance. We are still only beginning to realize the full importance and usefulness of fractal geometry.
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Fractals in Finance Finance played a crucial role in the development of fractal theory. Fractals are used in finance to make predictions as to the risk involved for particular stocks.
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Why does it matter? How is the stock market associated with a fractal? Easily, if one looks at the market price action taking place on the monthly, weekly, daily and intra day charts where you will see the structure has a similar appearance. Followers of this approach have determined that market prices are highly random but with a trend. They claim that stock market success will happen only by following the trend.
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Applications of fractals One of the most useful applications of fractals and fractal geometry is in image compression. It is also one of the more controversial ideas. The basic concept behind fractal image compression is to take an image and express it as an iterated system of functions. The image can be quickly displayed, and at any magnification with infinite levels of fractal detail. The largest problem behind this idea is deriving the system of functions which describe an image.
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Fractals in Film Industry One of the more trivial applications of fractals is their visual effect. Not only do fractals have a stunning aesthetic value, that is, they are remarkably pleasing to the eye, but they also have a way to trick the mind. Fractals have been used commercially in the film industry, in films such as Star Wars and Star Trek. Fractal images are used as an alternative to costly elaborate sets to produce fantasy landscapes.
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Other Applications of Fractals As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include: Classification of histopathology slides in medicine Fractal landscape or Coastline complexity Enzyme/enzymology (Michaelis-Menten kinetics) Generation of new music Signal and image compression Creation of digital photographic enlargements Seismology Fractal in soil mechanics
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Computer and video game design, especiallycomputer graphics for organic environments andas part of procedural generationFractography and fracture mechanicsFractal antennas – Small size antennas usingfractal shapesSmall angle scattering theory of fractally roughsystemsT-shirts and other fashionGeneration of patterns for camouflage, such asMARPATDigital sundialTechnical analysis of price series (see Elliott waveprinciple)
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Applications of Fractals in C.Sc. fractal techniques for data analysis fractals and databases, data mining visualization and physical models automatic object classification fractal and multi-fractal texture characterization shape generation, rendering techniques and image synthesis 2D, 3D fractal interpolation image denoising and restoration image indexing, thumbnail images
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fractal still image and video compression,wavelet and fractal transforms,benchmarking, hardwarewatermarking, comparison with othertechniquesbiomedical applicationsengineering (mechanical & materials,automotive)fractal and compilers, VLSI designinternet traffic characterization andmodelingnon classical applications
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