Fractal GeometryFractal Geometry

Dr Helen McAneneyDr Helen McAneney

Centre for Public Health,Centre for Public Health,Queen’s University BelfastQueen’s University Belfast

This talk

Steven H Strogatz, 1994. 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

1 2

3 4

• 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 Snowflake3 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