Mathieu Ledru
January 21, 2007
25

# Fractals

January 21, 2007

## Transcript

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

2. Fractals in nature
Fractals? Where?
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3. The cauliflower
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4. The fern
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5. The blitz
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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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7. The lungs
Without that structure, the lungs would took 2.8m3
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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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9. Construction of figures having this
self-similarity property
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10. Von Koch snowflake (1904)
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11. Principle
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12. The triangle and carpetof Sierpinsky
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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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14. How do we name these shapes with
this self-similarity principe?
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15. Fractal
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In 1975, Benoit Mandelbrot who name it « fractal »

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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17. The fractal’s dimension
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Yes, certain objects are of non-integer dimension

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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19. Beauty
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About the beautiful images fractals generated by transformations.

20. The Mandelbrot’s ensemble (1981)
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21. Process
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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

22. Particularity
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We always find the original size when we zoom in!!!

23. What could we do with fractals?
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24. The virtual images
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On the computer, one can reveal virtual images of natural objects of a
great complexity and of one extraordinary resemblance.

25. In industry
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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.

26. In art
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27. Hello!

I Am Mathieu Ledru
You can contact me at @matyo91
@matyo91
Thanks!
Any questions?
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