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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).

- 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...

Source: ccl.northwestern.edu

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)

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)

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)

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.

Koch’s Snowflake

3 iterations

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

Iteration 1

Iteration 2

Iteration 3

Iteration 4

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

Iteration 1

Iteration 2

Iteration 3

Iteration 4

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

Iteration 1

Iteration 2

Iteration 3

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