fractal geometry dr helen mcaneney centre for public health, queen’s university belfast

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