Fractal Geometry
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Fractal Geometry. Dr Helen McAneney. Centre for Public Health, Queen’s University Belfast. This talk. Steven H Strogatz, 1994. Nonlinear Dynamics and Chaos: with applications to Physics, Biology, Chemistry and Engineering (Addison-Wesley). Fractals.

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Fractal Geometry

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Fractal geometry

Fractal Geometry

Dr Helen McAneney

Centre for Public Health,

Queen’s University Belfast


Fractal geometry

This talk


Fractal geometry

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


Fractals

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

Mandelbrot Set


Netlogo mandelbrot

Netlogo: Mandelbrot

Source: ccl.northwestern.edu


Interface

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

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

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

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

Netlogo: L-System Fractals

Koch’s Snowflake

3 iterations


Fractal geometry

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

First attempt!


Fractal square

Fractal Square?

Iteration 1


Fractal square1

Fractal Square?

Iteration 2


Fractal square2

Fractal Square?

Iteration 3


Fractal square3

Fractal Square?

Iteration 4


Fractal geometry

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

Fractal Square 2?

Iteration 1


Fractal square 21

Fractal Square 2?

Iteration 2


Fractal square 22

Fractal Square 2?

Iteration 3


Fractal square 23

Fractal Square 2?

Iteration 4


Fractal geometry

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

Fractal Hexagon?

Iteration 1


Fractal hexagon1

Fractal Hexagon?

Iteration 2


Fractal hexagon2

Fractal Hexagon?

Iteration 3


New code

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


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