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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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East seacoast 11 x 1km 10km
East seacoast • Seacoast line length k.n(k) • lim k→0 k.n(k) = D
West seacoast • lim k→0 k.n(k) =∞
Koch snowflake Niels Fabian Helge von Koch (25.1. 1870 – 11.3.1924 Stockholm)
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 →∞
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
Mathematical definition • Fractal is a shape with Hausdorf dimension different of geometrical dimension
Non-fractal shapes • Refining the gauge s-times • The number of segments increase sD –times • D is geometrical dimension
Dimension of Koch snowflake • Koch curve • 3 x refining => 4 x length • s= 3 => N = 4 • D = logN/logs = log4/log3 = 1.261895
Other Hausdorf dimensions • Sierpinski carpet 1,58 • Menger sponge 2,72 • Pean curve 2 • Sea coastline 1,02 – 1,25
Polynomical fractals • Polynomicalrecursiveformula • Kn+1 = f(kn) • Thesequencedepending on theorigin k0 • Coverges • Diverges • Oscillates
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