Fractal geometry

Lewis Richardson, Seacoast line length

East seacoast

10 km

11 x 1km

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 snowflakeNiels 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 carpetHole 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

ExamplesC Z0 Z1 Z2 Z3 Z4

0 + 0i 0,0 0,0 0,0 0,0 0,0 Conv

In M.S.

1+0i 0,0 1,0 2,0 5,0 26,0 Div.

Not in M.S.

-1+0i 0,0 -1,0 0,0 -1,0 0,0 Osc.

In M.S.

-2+0i 0,0 -2,0 -2,0 -2,0 -2,0 Conv.

In M.S.-2,0000000001+0i

0,0 … … Div.

Not in M.S.

Mandelbrot set