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Fractals @matyo91 A geometry from the nature 30-01-2007

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Fractals in nature Fractals? Where? @matyo91 30-01-2007

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The cauliflower @matyo91 30-01-2007

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The fern @matyo91 30-01-2007

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The blitz @matyo91 30-01-2007

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The fractals in the human’s body This cauliflower is fractal! Some elements of the body’s corp are the same! Let us prove it @matyo91 30-01-2007

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The lungs Without that structure, the lungs would took 2.8m3 @matyo91 30-01-2007

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All the objects we view have the same property A fragment of the object look as the object itself! This is call : Self-similarity Self-similarity @matyo91 @matyo91 30-01-2007

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Construction of figures having this self-similarity property @matyo91 30-01-2007

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Von Koch snowflake (1904) @matyo91 30-01-2007

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Principle @matyo91 30-01-2007

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The triangle and carpetof Sierpinsky @matyo91 30-01-2007

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Menger sponge The number of cubes increases by : 20^n. Where n is the number of iterations performed on the first cube @matyo91 30-01-2007

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How do we name these shapes with this self-similarity principe? @matyo91 30-01-2007

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Fractal @matyo91 30-01-2007 In 1975, Benoit Mandelbrot who name it « fractal »

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It has a fine structure at arbitrarily small scales It possess the self-similarity structure It can have a non-integer dimension property of a fractal’s object @matyo91 @matyo91 30-01-2007

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The fractal’s dimension @matyo91 30-01-2007 Yes, certain objects are of non-integer dimension

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That is to say 1 the side of the initial triangle. Then in the second stage, the side of the three triangles and we got a shape twice larger. d= log(3)/ log(2) = 1.58 
 For S(3), we got 
 d = log(27)/log(8) = 1.58 Dimension of the Sierpinsky’s triangle @matyo91 30-01-2007

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Beauty @matyo91 30-01-2007 About the beautiful images fractals generated by transformations.

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The Mandelbrot’s ensemble (1981) @matyo91 30-01-2007

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Process @matyo91 30-01-2007 z0=0 
 z n+1 = zn2 + c and c is a complex. For each pixel of the screen, we associate a value of c. 
 If zi has a module higher than 2, the sequence diverges and the pixel is drawn color i. 
 When the sequence does not diverge, the pixel in black is colored

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Particularity @matyo91 30-01-2007 We always find the original size when we zoom in!!!

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What could we do with fractals? @matyo91 30-01-2007

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The virtual images @matyo91 30-01-2007 On the computer, one can reveal virtual images of natural objects of a great complexity and of one extraordinary resemblance.

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In industry @matyo91 30-01-2007 They are in the origin of : new materials of insulation like polymers. Processes of recovery of oil by injection of fluids under pressure in the porous rocks Etc.

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In art @matyo91 30-01-2007

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Hello! I Am Mathieu Ledru You can contact me at @matyo91 @matyo91 Thanks! Any questions? 30-01-2007