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Fractal geometry

Fractal geometry. Lewis Richardson, Seacoast line length. East seacoast. 11 x 1km. 10 km. East seacoast. Seacoast line length k.n(k) lim k→0 k.n(k) = D. Weat seacoast. West seacoast. lim k→0 k.n(k) =∞. Self-similarity. Koch snowflake.

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Fractal geometry

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  1. Fractal geometry

  2. Lewis Richardson, Seacoast line length

  3. East seacoast 11 x 1km 10km

  4. East seacoast • Seacoast line length k.n(k) • lim k→0 k.n(k) = D

  5. Weat seacoast

  6. West seacoast • lim k→0 k.n(k) =∞

  7. Self-similarity

  8. Koch snowflake Niels Fabian Helge von Koch (25.1. 1870 – 11.3.1924 Stockholm)

  9. Length of Koch snowflake 3 4/3 * 3 = 4 4/3*4/3*3 = 5,33 (4/3)3*3=7,11 (4/3)n*3 →∞

  10. Sierpinski carpet

  11. Area of Sierpinski carpet Hole area 1/9 8/9 * 1/9 (8/9)2 * 1/9 (8/9)n * 1/9 Suma 1/9 * ∑(8/9)i = 1 Area of the carpet = 1 – hole area = 0

  12. Menger sponge

  13. Natural fractals

  14. Natural self-similarity

  15. Mathematical definition • Fractal is a shape with Hausdorf dimension different of geometrical dimension

  16. Non-fractal shapes • Refining the gauge s-times • The number of segments increase sD –times • D is geometrical dimension

  17. Dimension of Koch snowflake • Koch curve • 3 x refining => 4 x length • s= 3 => N = 4 • D = logN/logs = log4/log3 = 1.261895

  18. Other Hausdorf dimensions • Sierpinski carpet 1,58 • Menger sponge 2,72 • Pean curve 2 • Sea coastline 1,02 – 1,25

  19. Polynomical fractals • Polynomicalrecursiveformula • Kn+1 = f(kn) • Thesequencedepending on theorigin k0 • Coverges • Diverges • Oscillates

  20. Mandelbrot set

  21. Mandelbrot set • Part of complex plane • z0 = 0,zn+1 = zn2 + c • If for given c the sequence • Converges  c is in Mandelbrot set • Diverges  c is not in Mandelbrot set • Oscillates  c is in Mandelbrot set

  22. Examples

  23. Mandelbrot set

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