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Fractals

 Fractals

Mathieu Ledru

January 21, 2007
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  1. Fractals
    @matyo91
    A geometry from the nature
    30-01-2007

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

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  3. The cauliflower
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    30-01-2007

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  4. The fern
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    30-01-2007

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  5. The blitz
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    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
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    30-01-2007

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

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

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

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  11. Principle
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    30-01-2007

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

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

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  15. Fractal
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    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
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    30-01-2007

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

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

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  21. Process
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    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
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    30-01-2007
    We always find the original size when we zoom in!!!

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  23. What could we do with fractals?
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    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
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    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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