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

This talk

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

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...

Mandelbrot Set

Netlogo: Mandelbrot Source: ccl.northwestern.edu

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)

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)

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)

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.

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

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

First attempt!

Fractal Square? Iteration 1

Fractal Square? Iteration 2

Fractal Square? Iteration 3

Fractal Square? Iteration 4

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

Fractal Square 2? Iteration 1

Fractal Square 2? Iteration 2

Fractal Square 2? Iteration 3

Fractal Square 2? Iteration 4

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

Fractal Hexagon? Iteration 1

Fractal Hexagon? Iteration 2

Fractal Hexagon? Iteration 3

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