1. Fractal geometry

2. Lewis Richardson, Seacoast line length

3. East seacoast
11 x 1km
10 km

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 ( – Stockholm)

9. Length of Koch snowflake
/3 * 3 = /3*4/3*3 = 5, (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 =

18. Other Hausdorf dimensions
Sierpinski carpet 1,58
Menger sponge 2,72
Pean curve 2
Sea coastline
1,02 – 1,25

19. Polynomical fractals Polynomical recursive formula
Kn+1 = f(kn)
The sequence depending on the origin 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 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, i …

23. Mandelbrot set