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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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Presentation Transcript
slide1
Fractal Geometry

Dr Helen McAneney

Centre for Public Health,

Queen’s University Belfast

slide3
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...
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

slide15
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

fractal square
Fractal Square?

Iteration 1

fractal square1
Fractal Square?

Iteration 2

fractal square2
Fractal Square?

Iteration 3

fractal square3
Fractal Square?

Iteration 4

slide21
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

slide26
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

ad