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 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 →∞
Sierpinski carpet
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
Menger sponge
Natural fractals
Natural self-similarity
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 Polynomical recursive formula Kn+1 = f(kn) The sequence depending on the origin k0 Coverges Diverges Oscillates
Mandelbrot set
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
Examples C Z0 Z1 Z2 Z3 Z4 0 + 0i 0,0 Conv In M.S. 1+0i 1,0 2,0 5,0 26,0 Div. Not in M.S. -1+0i -1,0 Osc. -2+0i -2,0 Conv. -2,0000000001+0i …
Mandelbrot set