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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