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Exploring Fractal Geometry & Self-similarity in Natural and Mathematical Forms

Delve into the world of fractal geometry, from the coastline lengths to intricate Mandelbrot sets, unveiling self-similarity and Hausdorf dimensions. Learn about Koch snowflakes, Sierpinski carpets, Menger sponges, and more! Discover polynomical fractals, Mandelbrot sets, and the fascinating Juliova množina. Dive deep into the mathematics of fractals and their dazzling beauty.

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Exploring Fractal Geometry & Self-similarity in Natural and Mathematical Forms

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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

  24. Juliovy množiny

  25. Juliova množina pro dané komplexní číslo c • Pro každý bod komplexní roviny z počítám • z0 = z • Zn+1 = zn2 + c (stejný vzorec jako u Mandelbrotovy množiny) • Pokud posloupnost zn nejde do nekonečna, je bod z prvkem Juliovy množiny pro číslo c, • Tuto množinu značíme Jc

  26. Pozorování • Juliova množina Jc vypadá v okolí bodu 0 podobně jako Mandelbrotova množina v okolí bodu c • Pro body c uvnitř Mandelbrotovy množiny je (0,0) prvkem Juliovy množiny Jc aJuliova množina Jc souvislá • Pro body c vně Mandelbrotovy množiny je Juliova množina Jc nesouvislá, popřípadě prázdná.

  27. Pozorování Pro body c „hodně uvnitř“ Mandelbrotovy množiny je Juliova množina Jc nezajímavý souvislý útvar.

  28. Pozorování Pro body c „hodně vně“ Mandelbrotovy množiny tvoří Juliovu množinu Jc několik izolovaných bodů

  29. Pozorování „Nejzajímavější“ Juliovy množiny vzniknou z bodů, které leží poblíž hranice Mandelbrotovy množiny, ať již zevnitř

  30. Nebo zvenku

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