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Exploring the Wonderful World of MATHEMATICS SECRET OF FRACTALS Dr. Farhana Shaheen Yanbu University College, KSA
WHAT ARE FRACTALS?• For centuries, mathematicians rejected complex figures, leaving them under a single description: “formless”.• For centuries, geometry was unable to describe trees, landscapes, clouds, and coastlines. However, in the late 1970’s a revolution of our perception of the world was brought by the work of Benoit Mandelbrot who introduced FRACTALS.
“Fractua” means Irregular• Fractals are geometric figures like circles, squares, triangles, etc., but having special properties. They are usually associated with irregular geometric objects, that look the same no matter at what scale they are viewed at.• A fractal is an object in which the individual parts are similar to the whole.
Fractals exhibit self-similarity• Fractals have the property of self- similarity, generated by iterations, which means that various copies of an object can be found in the original object at smaller size scales.• The detail continues for many magnifications - - like an endless nesting of Russian dolls within dolls.
What exactly 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.
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.
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. • Mandelbrot Set
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.
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.
Difference between the Julia set and 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.
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 known as strange attractors.
Objects in NatureMany objects in nature aren’t formed of squares ortriangles, but of more complicated geometric figures.e.g. trees, ferns, clouds, mountains etc. are shapedlike fractals. Other examples include snowflakes, crystals, lightning, river networks, caulifloweror broccoli, and systems of blood vessels andpulmonary vessels. Coastlines may also beconsidered as fractals in nature.
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.
Examples of Fractals in Nature• A cauliflower is a perfect example of a fractal where each element is a perfect recreation of the whole.
A naturally occurring Cauliflower Fractal• 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.
Similarity between fractals and objects 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.
Computer-generated Fractal patterns• These days computer-generated fractal patterns are everywhere. From squiggly designs on computer art posters to illustrations in most of physics journals, interest continues to grow among scientists and, rather surprisingly, artists and designers.
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?
The Sierpinski Triangle• Cut out the triangle in the center.• Step Two• Draw another equilateral triangle with sides of 4 triangle lengths each. Connect the midpoints of the sides and cut out the triangle in the center as before.
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.
Natural fractal pattern - air displacing a vacuum formed by pulling two glue-covered acrylic sheets apart.
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.
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.
Applications of Fractals in Science• Fractals have a variety of applications in science because its property of self similarity exists everywhere. They can be used to model plants, blood vessels, nerves, explosions, clouds, mountains, turbulence, et c. Fractal geometry models natural objects more closely than does other geometries.• Engineers have begun designing and constructing fractals in order to solve practical engineering problems. Fractals are also used in computer graphics and even in composing music.
Application of Fractals and Chaos is in Music• Some music, including that of Bach 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.
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.
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.
Real-Life Relevance And Importance of Fractals and Fractal Geometry – 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.
Fractals in Finance and Risk Fractals in Finance
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.
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.
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.
Fractals in Film IndustryOne of the more trivial applications of fractals is theirvisual effect. Not only do fractals have a stunningaesthetic value, that is, they are remarkably pleasingto the eye, but they also have a way to trick themind. Fractals have been used commercially in thefilm industry, in films such as Star Wars and Star Trek.Fractal images are used as an alternative to costlyelaborate sets to produce fantasy landscapes.
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
• Computer and video game design, especially computer graphics for organic environments and as part of procedural generation• Fractography and fracture mechanics• Fractal antennas – Small size antennas using fractal shapes• Small angle scattering theory of fractally rough systems• T-shirts and other fashion• Generation of patterns for camouflage, such as MARPAT• Digital sundial• Technical analysis of price series (see Elliott wave principle)
Applications of Fractals in Computer Science• 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
• fractal still image and video compression, wavelet and fractal transforms, benchmarking, hardware• watermarking, comparison with other techniques• biomedical applications• engineering (mechanical & materials, automotive)• fractal and compilers, VLSI design• internet traffic characterization and modeling• non classical applications
• Dear Students and Colleagues,This is not THE END…This is just the beginning…… to start exploring…… the Wonderful World of Mathematics Thank Youhttp://www.youtube.com/watch?feature=player_e mbedded&v=tPef_7QPcXI