Fractals

Transcript

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

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

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3. The cauliflower
   @matyo91
   30-01-2007
   

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4. The fern
   @matyo91
   30-01-2007
   

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5. The blitz
   @matyo91
   30-01-2007
   

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

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

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

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11. Principle
    @matyo91
    30-01-2007
    

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

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

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

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16. 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
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    @matyo91
    30-01-2007
    

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

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

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

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

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

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24. 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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25. 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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26. In art
    @matyo91
    30-01-2007
    

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

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