A Fractal View of the World

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A Fractal View of the World

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented to the
• CHAOS AND COMPLEX SYSTEMS SEMINAR
• in Madison, Wisconsin
• on February 17, 2009

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Fractals

• Geometrical objects generally with non-integer
  dimension
• Self-similarity (contains infinite copies of
  itself)
• Structure on all scales (detail persists when
  zoomed arbitrarily)

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Cantor Set
D log 2 / log 3 0.6309
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Cantor Curtains
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Fractal Curves
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Weisstrass Function
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Fractal Trees
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Lindenmayer Systems
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Fractal Gaskets
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Natural Fractals
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Fractal Dimension
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Power Laws (Zipf)
Size of Power Outages
Words in English Text
Earthquake Magnitudes
Internet Document Accesses
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Other Examples of Power Laws

• Populations of cities
• Size of moon craters
• Size of solar flares
• Size of computer files
• Casualties in wars
• Occurrence of personal names
• Number of papers scientists write
• Number of citation received
• Sales of books, music,
• Individual wealth, personal income
• Many others

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Stretching and Folding
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Strange Attractors
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Strange Attractor Icon
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Iterated Function System
Heads Tails
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Iterated Function System Icon
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Generalized Julia Set
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Criterion after at least 100 iterations but
less than 1000 iterations for x0 y0 0
y0
-1
-1 x0
1
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Human Evaluations
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Aesthetic Evaluation
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References

• http//sprott.physics.wisc.edu/
  lectures/fractals.ppt (this talk)
• http//sprott.physics.wisc.edu/ fractals.htm (my
  fractal gallery)
• http//sprott.physics.wisc.edu/chaostsa/ (my
  chaos textbook)
• sprott_at_physics.wisc.edu (email me)