Fractal Geometry

Fractal Geometry Math Carla Moreno 6th period

The concept… Fractal Geometry , also called the “true geometry of nature” could be explained as: non-triangular or squared shaped figures, but as more complex ones. Fractals are used to explain concrete objects. For example : in regular objects as in nature, we need fractal geometry. Fractal geometry deals with non-integer dimensions. Zero dimensional point is only a point. One is a Line or a curve , two dimensional figures could be plane figures as circles , triangles or squares. Three dimensional solid figures are solids such as cubes, pyramids, cylinders and spheres. (three dimensional figures). Which are used just for a model of reality. Unlike the other dimensions, the fourth one is located in the real world, in the one we live.

Continuation We are actually living in the fourth dimension of the “space-time continuum “ “Fractals are an open system where everything is related to everything else. “ Before fractal geometry there wasEuclidian geometry which was used thousands of years ago but it only stated artificial realities, such as: the fist , second and third dimension. These dimensions are imaginary, only the fourth is real. These caused great advance in math. Math now, is a complete subject , separated completely by physics (science) its sister subject for centuries. Fractals were discovered in the decade of the 1970’s.

What are images? Images are geometric figures in irregular shape in form of pattern that repeats itself with similarity; sometimes even with exactitude in an endless manner having a bigger image, being repeated into smaller ones, and smaller and each time smaller, until they are practically invisible to the human eye.

Example of images: Julia set with cousines Julia set Mandelbrot set A Mandelbrot shape wich apears in seahorse valley after magnification Sierpinski’s triangle

Some images

Uses of fractal Geometry: Medicine (Science) Fractals are found in the body. Fractals can help us to understand better the human body. The best known example are the arteries and veins in mammalian vascular systems. “This area of biological research is just beginning. Chaotician Michael McGuire refers to recent discoveries in brain research which suggests that a fractal structure based on hexagons may be how the receptive fields of the visual cortex are organized. The smallest hexagons correspond to the cells of the retina and perception of fine details, the larger hexagons organize the underlying layers to recognize progressively coarse detail.”

Entratainment industry Fractal geometry is used also in entertainment, (technology) industry . Fractal geometry can be used today in movie making , principally in the especial effects and animation in three dimensional cartoons, like for example: Jimmy Neutron and The ice age, which where only created with fractals. Thanks to these, we are leading to a new generation of cartoons and television series.

Music industry Fractal Geometry can also be used in music industry. In order to a song , to have a certain rhythm, it has to repeat its self over and over, with out losing the original beat, therefore, fractal geometry is used for music. Also for the “music waves” and for the images, fractal geometry is used. Midi files, which are the ones for the use on internet web pages, are especially formed with fractals and Iteration because they never end, they are design to keep up on going and going, until the user closes the window.

Philosophy and …nature Philosophy are mainly referring at the study of men, how they think and to think about explanations for things. Well guess, what know fractal geometry is also affecting this because fractal geometry is the why about everything in nature, the way it is. It’s mainly telling everything with math. Why flower are like that or why snowflakes are awesome, because there’s never another, exactly the same , they have infinite patterns.

Science: space, meteorology Space is an excellent example to describe Fractals. What we’ve noticed about space we can state that are fractals. An image of space can be revealed as a fractal image. We, human don’t even know were the universe ends. Therefore, we can state it as a fractal, fractals many times don’t end, they just repeat, just as space. Meteorologists apply fractals, to comprehend the form, shape and even the composition of natural bodies such are clouds or as already said, snowflakes.

Biography: Benoit Mandelbrot BenoitMandelbrot. He is responsible for the actual interest in fractal geometry. He showed how fractals can happen in many different places in both: mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924. His family had a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

Continuation… Mandelbrot's family emigrated to France in 1936. His uncle Szolem Mandelbrojt, who was Professor of Mathematics at the College de France and successor of Hadamard in this post, took responsibility for his education. Mandelbrot attended “Lycée Rolin” in Paris up till the start of the second World War. This was a very difficult period during Mandelbrot’s life, who feared for his life on various occasions. Thanks to his unconventional education , Mandelbrot thought in other ways that would be difficult for someone that had an conventional education.

Then Mandelbrot went to study at Lyon , Paris. There, Paul Lévy teached Mandelbrot and became one strong influence for him. Then he went to Princeton for the advance study of technology, and he was sponsored by John von Neumann. Mandelbrot went back to France in 1955 and worked at the Centre National de la Recherche Scientific. There, he married Aliette Kagan, he didn’t stay there too long before returning to the United States. He was unhappy. In 1945 Mandelbrot's uncle had introduced him to Julia’s important 1918 paper claiming.

Mandelbrot was asked to work for the IBM company (in the watson reserch center), there he had a wonderful space of work, and he like it , besides there he could do his reserch. Mandelbrot met Julia, and later he published Julia's work . Jualia’s work is now known as the most beautiful fractals of today. His work was printed in his book “Les objets fractals, forn, hasard et dimension” (1975) and more complete in The fractal geometry of nature in 1982. On 23 June 1999 Mandelbrot got the Honorary Degree of Doctor of Science from the University of St Andrews.

While he was an IBM fellow, he was a professor of mathematics at Harvard University and ´Ecole Polytechnique, professor of engineering at Yale, Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. “Mandelbrot has received numerous honors and prizes in recognition of his remarkable achievements. Just for example, in 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honored with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993”.

Mandelbrot also discovered Madelbrot set. Which is a simple formula: z -> z^2 + c , even with computers, the formula couldn't had been discovered without him. With these formula , the world has been done a great change. Benoit Mandelbrot, is now an IBM scientist and Professor of Mathematics at Yale.

Iteration, what is it? How does it relate? Many sets, principally the Mandelbrot set are generated by iteration. Iteration means to repeat a process over and over again. In mathematics, this process is most often the use of a mathematical function. For example: “for the Mandelbrot set, the function involved is the simplest nonlinear function imaginable, namely x2 + c, where c is a constant. ”The answer has been got from an specific and repetitive order of operations. This formula at the end, it tends to Infinity, therefore, it has to be a lot with fractals. Like as I mentioned at the beginning: “Fractals are endless repetitive patterns .” so, Iteration makes up the repetitive part of fractals, besides , Iteration makes fractals more precise.

How are imaginary numbers related? Although “imaginary numbers” are not imaginary; in fact they a very real, these numbes are very useful for the fractal geometry. In fractal geometry people coulnd’t get the fractals right because real numbers are just “big”. With imaginary numbers people can’t state how many litters are in a bottle of water or how meny kilos are in a bag. Imaginary numbers are used to calculate things that can´t be calculated with “real” numbers. Imaginary numbers are related to fractals, in the way that fractals can’t be calculate with real numbers, therefore we need the “imaginary numbers” with out them , fractal geometry wouldn’t be possible.

Conclusions: Fractals are extremely important. Fractals are just another way to see life. Everything has to do with them , even the most little thing in the world that I can think of. Fractals are the future of humans, fractals are leading humans to a complete new generation. There is no doubt that thanks to Mandelbrot's discovery , the human specie is going to advance way too much. I liked working on this project more than other I had worked on in math. I think this project is very original, and different from the others. This has a clear point and I really learned about it because before I didn’t had a clue about what fractal geometry or a fractal was. I practically learned all I wrote. It really caught my attention. This helped me to understand imaginary numbers better, because now I can say that imaginary numbers are actually non-imaginary numbers that complement a very important part of our world.

Bibliography: • http://home.inreach.com/kfarrell/fractals.html • http://math.rice.edu/~lanius/fractals/WHY/ • http://srd.yahoo.com/S=2766679/K=fractal+geometry+%2b+its+uses/v=2/l=WS1/R=3/H=0/*-http://www.fractalwisdom.com/FractalWisdom/fractal.html • http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html • http://www.io.com/~mgscheue/chaos/iteration.html • http://hilbert.dartmouth.edu/~doyle/docs/icos/icos/icos.html • http://hilbert.dartmouth.edu/~doyle/docs/icos/icos/icos.html • http://astronomy.swin.edu.au/~pbourke/fractals/ • http://srd.yahoo.com/S=2766679/K=imaginary+numbers/v=2/l=WS1/R=1/SS=8865752/H=0/*-http://www.math.toronto.edu/mathnet/answers/imaginary.html

Fractal Geometry