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math 1-roque

gerron sagarang

on 30 September 2012

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Transcript of fractals

FRACTAL GEOMETRY What Are Fractals? when the word fractal is mentioned, you immediately think of the stunning pictures you have seen that were called fractals. But just what exactly is a fractal? Basically, it is a rough geometric figure that has two properties First, most magnified images of fractals are essentially indistinguishable from the unmagnified version. This property of invariance under a change of scale if called self-similiarity Second, fractals have fractal dimensions The word fractal was invented by Benoit Mandelbrot, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means to break to create irregular fragments. It is therefore sensible and how appropriate for our needs! - that, in addition to fragmented, fractus should also mean irregular, both meanings being preserved in fragment." Real-life Application of Fractals: Nature:
Trees, Clouds, Snowflakes Astronomy:
Cosmologists usually assume that matter is spread uniformly across space.
Astronomers agree with that assumption on "small" scales, but most of them think that the universe is smooth at very large scales.
A dissident group of scientists claims that the structure of the universe is fractal at all scales.
Based on the concepts and methods of modern Statistical Physics, galaxy structures are highly irregular and self-similar.
At present, cosmologists need more data about the matter distribution in the universe to prove (or not) that we are living in a fractal universe. Computer Science (Data and Image Compression):
Data Compression: Microsoft’s Encarta Encyclopedia
The basic concept behind fractal image compression is to take an image and express it as an iterated function system (IFS).
Images are compressed much more than by usual ways (i.e. JPEG or GIF file formats)
The image can be quickly displayed, and at any magnification with infinite levels of fractal detail.
When the picture is enlarged, there is no pixelization.
The largest problem behind this idea is deriving the system of functions which describe an image. Fluid Mechanics:
Turbulence flows
Porous media Telecommunications:
Fractenna: A company which sells fractal-shaped antennae that reduce greatly the size and the weight of the antennas
The fractal parts produces 'fractal loading' and makes the antenna smaller for a given frequency of use Surface Physics
Fractals are used to describe the roughness of surfaces.
A rough surface is characterized by a combination of two different fractals. Medicine:
Biosensor interactions can be studied by using fractals. Art (Visual Effects):
Stunning aesthetic value, pleasing to the eye, mind tricks
Has been used commercially in the film industry (Star Wars, Star Trek).
An alternative to costly elaborate sets to produce fantasy. 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 lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithims -- a set of instructions on how to create a fractal. The world as we know it is made up of objects which exist in integer dimensions, single dimensional points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three dimensional solid objects such as spheres and cubes. However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists. The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension. The Mandelbrot set Named after Benoit Mandelbrot, The Mandelbrot set is one of the most famous fractals in existance. 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 iterating z=z2+c does not diverge.
To generated the Mandlebrot set graphically, the computer screen becomes the complex plane. Each point on the plain is tested into the equation z=z2+c. If the iterated z stayed withen a given boundry forever, convergence, the point is inside the set and the point is plotted black. If the iteration went of control, divergence, the point was plotted in a color with respect to how quickly it escaped. 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.

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. The 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. 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.

After closer examinition of the process used to generate the Sierpinski Triangle and the image produced by this process, we realize that the magnification factor is two. With each magnification, there are three divisions of the triangle. The Sierpinski Triangle The Koch Snowflake As would be expected, the Koch Snowflake is generated in very much the same way as the Koch Curve. The only variation is that, rather than using a line of unit length as the intial figure, an equilater triangle is used. It is iterated in the same way as the Koch Curve. The length of the resulting figure tends to infinity as the length of the side of each new triangle goes to zero. Iterated an infinite number of times, the Koch Snowflake, like the Koch Curve, has absolutely no straight lines in it. This fractal, if magnified three times in any area, also displays the property of self-similiarity. The Koch Curve The Koch curve was named after Helge von Koch in 1904. The generation of this fractal is simple. We begin with a straight line of unit length and divide it into three equally sized parts. The middle section is replaced with and equilateral triangle and its base is removed. After one iterations, the length is increased by four-thirds. As this process is repeated, the length of the figure tends to infinity as the length of the side of each new triangle goes to zero. Assuming this could be iterated an infinite number of times, the result would be a figure which is infinitely wiggly, having no straight lines The Cantor set The Cantor set is a good example of an elementary fractal.The set is generated by the iteration of a single operation on a line of unit lenght. With each iteration, the middle third from each lines segment of the previous set is simply removed. As the number of iterations increases, the number of seperate line segments tends to infinity while the length of each segment approaches zero. Under magnification, its structure is essentially indiguishable from the whole, making it self-similiar. END ^_^ http://library.thinkquest.org/3493/frames/fractal.html

youtube:Mandelbrot Fractal Zoom: Bach, Fractals, and The Art of Fugue (HD) Members:
Rodriguez, Mark Kenneth
Sarmiento, Cathe
Sagarang, Gerron
Salcedo, Anne Clare
Sangalang, Vanessa
Santos, Anna
Sanchez, Miki
Sanchez, kate
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