Fractals

1
Fractals
2
Topics

• Why fractals?
• Fractal Dimension
• Mandelbrot Set
• Fractal Applications
• Discussion

3
Fractals?

• Fractal geometry of nature and chaos
• The middle ground of organized or orderly
  geometric chaos (Mandelbrot 1989)
• Derived partly because of the difficulty in
  analyzing spatial forms and processes (Lam and
  Quattrochi 1992)
• Pack an infinite amount of length into a small
  finite area, similar to the coastline scale issue
  (Flake 1998).
• A 1 inch Koch curve at 100 iterations would
  stretch around the earth 40,000 times.

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Fractal Dimension
1 Dimension (line) 221
2 Dimensions (square) 422
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Fractal Dimension
3 Dimensions (cube) 823
Number of copies made by doubling
2dimension
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Fractal Dimension
Fractal Dimension (Sierpinski Triangle) 32?
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Calculating Dimension with Log

• For a, the original segment is ¼ of the line
  length.
• For b, the original square is ½ of its side.
• For c, the original curve is 1/3 of the length of
  the iterated curve.

(Lam and Quattrochi 1992)
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Fractal Dimension

• In fractal geometry, fractal curves are between
  dimension 1 and 2, and fractal surfaces are
  between dimension 2 and 3.
• Coastlines typically are about fractal dimension
  of 1.2, and their relief dimension is about 2.2.
• Fractal dimensions of 1.5 and 2.5 are too large,
  or irregular, for modeling earth features.
• Fractal dimension is an indicator of complexity
  (Zhou and Lam 2005)
• Image Characterization and Measurement System
  (ICAMS) for dimension (Ibid)

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The Mandelbrot Set

• Xt1 xt2 c, where c some imaginary number.
• Ex. for c i, x0 0
• X1 0 i
• X2 i2 i -1 i
• X3 (-1 i)2 i i2 2i 1 i -i ..
• Points within the Mandelbrot
• set are black.

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Fractal Applications

• Categorization of phenomena using dimension
  similarity.
• Simulation (coast lines, stream patterns,
  surfaces and terrain, etc.)
• Simulated landscapes have been used to model
  habitat destruction (Malanson 2002).
• Art

11
Fractal Art
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Fractals and Hollywood

• Star Trek Genesis Effect
• Star Wars Death Star
• The Last Starfighter Landscapes
• The Perfect Storm
• Apollo 13
• Titanic

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Discussion

• Are fractals valuable to us now?
• Can you see any applications not discussed which
  might be useful?
• What does the future of fractals in geography
  look like?

14
Sources

• Flake, Gary W. (1998). The Computational Beauty
  of Nature Computer Explorations of Fractals,
  Chaos, Complex Systems, and Adaptation.
  Cambridge, The MIT Press.
• Lam, Nina S.-N., and Quattrochi, Dale A. (1992).
  On the Issues of Scale, Resolution, and Fractal
  Analysis in the Mapping Sciences. Professional
  Geographer, 44(1).
• Malanson, George P. (2002). Extinction-Debt
  Trajectories and Spatial Patterns of Habitat
  Destruction. Annals of the Association of
  American Geographers, 92(2).
• Mandelbrot, Benoit B. (1983). The Fractal
  Geometry of Nature. New York, W. H. Freeman and
  Company.
• Zhou, Guiyun, and Lam, Nina S.-N. (2005). A
  Comparison of Fractal Dimension Estimators Based
  on Multiple Surface Generation Algorithms.
  Computers and Geosciences, 31.