Fractals

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Fractals

• Joceline Lega
• Department of Mathematics
• University of Arizona

2
Outline

• Mathematical fractals
• Julia sets
• Self-similarity
• Fractal dimension
• Diffusion-limited aggregation
• Fractals and self-similar objects in nature
• Fractals in man-made constructs
• Aesthetical properties of fractals
• Conclusion

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Julia sets

• The pictures on the left represent a (rotated)
  Julia set.
• Consider iterations of the transformation defined
  in the plane bywhere cr and ci are parameters.
• Example cr 0, ci 1

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Julia sets (continued)

• More concisely,
  .
• Choose a pair of parameters (cr,ci).
• If iterates of the point z0 with coordinates
  (x0,y0) remain bounded, then this point is part
  of the corresponding Julia set.
• If not, one can use a color scheme to indicate
  how fast iterates of (x0,y0) go to infinity.
• Here, the darker the tone of red, the faster
  iterates go to infinity.

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Douady's rabbit fractal

• The fractal shown below is the Julia set for cr
  -0.123, ci 0.745

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Self-similarity
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Self-similarity
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Self-similarity
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Self-similarity
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Self-similarity
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Siegel disk fractal

• The fractal shown below is the Julia set for cr
  c-0.391, ci -0.587

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Julia sets

• As one varies (cr,ci), the complexity of the
  corresponding Julia set changes as well.
• The movie showsthe Julia sets for ci 0.534,
  and crvarying between0.4 and 0.6.
• One would like tomeasure the levelof
  complexity of each Julia set.

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Fractal dimension

• Consider an object on the plane and cover it
  with squares of side length L.
• Call N(L) the number of squares needed.
• For a smooth curve, N(L) 1/L L-1 and
  .
• For a fractal curve, the fractal dimension
  is such that df gt 1.

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Fractal dimension of Julia sets

• D. Ruelle showed that the fractal dimension of
  the Julia set of the quadratic map iswhere c
  cr i ci , c2cr2ci2 .
• In the movie shown before, c2cr2 0.5342,
  with 0.4 cr 0.6.
• The fractal dimension measures the level of
  complexity of the fractal.

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Diffusion-limited aggregation

• Place a seed (black dot) in the plane.
• Release particleswhich perform arandom walk.
• If the particletouches, theseed, it sticks
  toit and a new particleis released.
• If a particle wandersoff the box, it is
  eliminated and a new particle is released.

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Conclusions

• Fractals are mathematical objects which are
  self-similar at all scales.
• One way of characterizing them is to measure
  their fractal dimension.
• Many objects found in nature are self-similar,
  and the fractal dimension of landscape features
  is close to 1.3.
• The human eye appears to be tuned so that objects
  with a fractal dimension close to 1.3 are
  aesthetically pleasing.
• Such ideas can be used to create fractal-based
  virtual landscapes.

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Virtual landscape
Created with Terragen http//www.planetside.co.uk
/terragen/
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Examples of research projects

• Exploring Julia sets
• Complex variables (MATH 421, 424)
• Proof course (MATH 322)
• MATLAB
• Understanding DLA
• Probability (MATH 464)
• MATLAB
• Applications to bacterial colonies and other
  growth models
• ODEs (MATH 454) and PDEs (MATH 456)
• MATLAB
• Numerical Analysis (MATH 475)
• Exploring self-similarity in nature and in the
  laboratory.

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Homework problems

• Julia sets
• How would you set up a computer program to plot
  Julia sets?
• Use MATLAB to set up such a code.
• DLA
• Design a computer code that simulates the random
  walk of a particle. The simulation should stop if
  the particle leaves the box or reaches a
  pre-defined cluster inside the box.
• Program this in MATLAB.