Fractal Geometry

1. Fractal Geometry Dr Helen McAneney Centre for Public Health, Queen’s University Belfast

2. This talk

3. Steven H Strogatz, 1994. Nonlinear Dynamics and Chaos: with applications to Physics, Biology, Chemistry and Engineering (Addison-Wesley).

4. Fractals • Term coined by Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured.“ • Self-similarity, i.e. look the same at different magnifications • Mathematics: A fractal is based on an iterative equation • Mandelbrot set • Julia Set • Fractal fern leaf • Approx. natural examples • clouds, mountain ranges, lightning bolts, coastlines, snow flakes, cauliflower, broccoli, blood vessels...

5. Mandelbrot Set

9. Netlogo: Mandelbrot Source: ccl.northwestern.edu

10. Interface set z-real c-real + (rmult z-real z-imaginary z-real z-imaginary) set z-imaginary c-imaginary + (imult temp-z-real z-imaginary temp-z-real z-imaginary)

11. Extension1 set z-real c-real - (rmult z-real z-imaginary z-real z-imaginary) set z-imaginary c-imaginary - (imult temp-z-real z-imaginary temp-z-real z-imaginary)

12. Extension2 set z-real c-real - (rmult z-real z-imaginary z-real z-imaginary) set z-imaginary c-imaginary + (imult temp-z-real z-imaginary temp-z-real z-imaginary)

13. 1 2 3 4 Koch Snowflake • With every iteration, the perimeter of this shape increases by one third of the previous length. • The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite.

14. Netlogo: L-System Fractals Koch’s Snowflake 3 iterations

15. Code to kochSnowflake ask turtles [set new? false pd] ifelse ticks = 0 [repeat 3 [ t ahead len l 60 t ahead len r 120 t ahead len l 60 t ahead len r 120 ] ] [t ahead len l 60 t ahead len r 120 t ahead len l 60 t ahead len r 120 ] set len (len / 3) d end

16. First attempt!

17. Fractal Square? Iteration 1

18. Fractal Square? Iteration 2

19. Fractal Square? Iteration 3

20. Fractal Square? Iteration 4

21. Code to kochSnowflakenew2 ask turtles [set new? false pd] ifelse ticks = 0 [repeat 4 [t ahead len l 90 t ahead len r 90 t ahead len r 90 t ahead len l 90 t ahead len r 90 ] ] [t ahead len l 90 t ahead len r 90 t ahead len r 90 t ahead len l 90 t ahead len r 90 ] set len (len / 3) d end

22. Fractal Square 2? Iteration 1

23. Fractal Square 2? Iteration 2

24. Fractal Square 2? Iteration 3

25. Fractal Square 2? Iteration 4

26. Code to kochSnowflakenew2 ask turtles [set new? false pd] ifelse ticks = 0 [repeat 4 [t ahead len r 90 t ahead len l 90 t ahead len l 90 t ahead len r 90 t ahead len r 90 ] ] [t ahead len r 90 t ahead len l 90 t ahead len l 90 t ahead len r 90 t ahead len r 90 ] set len (len / 3) d end

27. Fractal Hexagon? Iteration 1

28. Fractal Hexagon? Iteration 2

29. Fractal Hexagon? Iteration 3

30. New Code Changed heading to -30 to kochSnowflakeNEW ask turtles [set new? false pd] ifelse ticks = 0 [ repeat 6 [ t ahead len l 60 t ahead len r 60 t ahead len r 60 t ahead len l 60 t ahead len r 60 ] ] [ t ahead len l 60 t ahead len r 60 t ahead len r 60 t ahead len l 60 t ahead len r 60 ] set len (len / 4) d end