Fractal geometry Lewis Richardson Seacoast line length East

Fractal geometry

Lewis Richardson, Seacoast line length

East seacoast 11 x 1 km 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) =∞

Soběpodobnost

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)n*3 →∞ 4/3*3 = 5, 33 (4/3) 3*3=7, 11

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 s. D – times • D is geometrical dimension

Dimension of Koch snowflake • Koch curve – 3 x refining => 4 x length – s = 3 => N = 4 – D = log. N/logs = log 4/log 3 = 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 k 0 – Coverges – Diverges – Oscillates

Mandelbrotova množina

Mandelbrot set • Part of complex plane • z 0 = 0, zn+1 = zn 2 + 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 Z 0 Z 1 Z 2 Z 3 Z 4 0 + 0 i 0, 0 0, 0 1+0 i 0, 0 1, 0 2, 0 5, 0 26, 0 -1+0 i 0, 0 -1, 0 0, 0 -2+0 i 0, 0 -2, 0 -2, 000001+0 i 0, 0 … … Conv In M. S. Div. Not in M. S. Osc. In M. S. Conv. In M. S. Div. Not in M. S.

Mandelbrot set