SNA of local PH sector

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

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Mandelbrot Set
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(No Transcript)
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(No Transcript)
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Netlogo Mandelbrot
Source ccl.northwestern.edu
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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)
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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)

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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)

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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.

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Netlogo L-System Fractals
Kochs Snowflake 3 iterations
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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

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First attempt!
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Fractal Square?
Iteration 1
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Fractal Square?
Iteration 2
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Fractal Square?
Iteration 3
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Fractal Square?
Iteration 4
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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

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Fractal Square 2?
Iteration 1
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Fractal Square 2?
Iteration 2
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Fractal Square 2?
Iteration 3
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Fractal Square 2?
Iteration 4
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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

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Fractal Hexagon?
Iteration 1
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Fractal Hexagon?
Iteration 2
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Fractal Hexagon?
Iteration 3
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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