Fractals

Transcript

1. Fractals @matyo91 A geometry from the nature 30-01-2007
2. Fractals in nature Fractals? Where? @matyo91 30-01-2007
3. The cauliflower @matyo91 30-01-2007
4. The fern @matyo91 30-01-2007
5. The blitz @matyo91 30-01-2007
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
7. The lungs Without that structure, the lungs would took 2.8m3 @matyo91 30-01-2007
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
9. Construction of figures having this self-similarity property @matyo91 30-01-2007
10. Von Koch snowflake (1904) @matyo91 30-01-2007
11. Principle @matyo91 30-01-2007
12. The triangle and carpetof Sierpinsky @matyo91 30-01-2007
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
14. How do we name these shapes with this self-similarity principe? @matyo91 30-01-2007
15. Fractal @matyo91 30-01-2007 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 @matyo91 @matyo91 30-01-2007
17. The fractal’s dimension @matyo91 30-01-2007 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 @matyo91 30-01-2007
19. Beauty @matyo91 30-01-2007 About the beautiful images fractals generated by transformations.
20. The Mandelbrot’s ensemble (1981) @matyo91 30-01-2007
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
22. Particularity @matyo91 30-01-2007 We always find the original size when we zoom in!!!
23. What could we do with fractals? @matyo91 30-01-2007
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.
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.
26. In art @matyo91 30-01-2007
27. Hello! I Am Mathieu Ledru You can contact me at @matyo91 @matyo91 Thanks! Any questions? 30-01-2007