FUN With Fractals and Chaos!

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FUN With Fractals and Chaos!
Team Project 3 Eric Astor, Christine Boone,
Eugene Astrakan, Benjamin Wieder, Stephanie Mok,
Matthew Zegarek, Alexandra Konings, John Cobb,
Scott Weingart, Dhruva Chandramohan Advisor Dr.
Paul Victor Quinn Sr. Assistant Karl Strohmaier
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What is a Fractal?

• Exhibits self-similarity

• Unique dimensionality

• Based on recursive algorithms

• Scale independent

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Sections

• Fractal Dimensions
• Sierpinski n-gons
• Nature Fractals
• Physics Fractals
• Mandelbrot/Julia Set
• Chaos Theory
• Bouncing Ball Model

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What is the Fractal Dimension?

• Non-integer dimension in which various patterns
  exist
• Characteristics of a fractal can be determined by
  calculating the value of its dimension

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How is the value of the fractal dimension
calculated?

• Box Counting Method
• Count number of occupied boxes
• Plot ln(occupied boxes) vs. ln(1/boxes per side)
• Slope gives fractal dimension

Fractal Dimension
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Geometric Formula

• Dimension ln (self similar pieces)
• ln (magnification)

D ln 4 ln 2

• Smaller magnification improves accuracy

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Sierpinski Fractals

• Named for Polish mathematician Waclaw Sierpinski
• Involve basic geometric polygons

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Sierpinski Triangle
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Sierpinski Triangle
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Sierpinski Square
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Sierpinski Carpet
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Sierpinski Carpet
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Other Sierpinski Polygons
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The SURACE 17-gon
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Sierpinski Chaos Game
Vertex 1
Midpoint
New Starting Point
Starting Point
Vertex 2
Vertex 3
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Sierpinski Chaos Game

• 100 pts

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Sierpinski Chaos Game

• 1000 pts

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Sierpinski Chaos Game

• 5000 pts

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Sierpinski Chaos Game

• 20000 pts

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Sierpinski Triangle Data
Fractal dimension 1.8175
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Fractals in Nature
Fractal Fern

• Initial X,Y starting point randomly chosen
• Probabilities indicate equation
• Plot X,Y coordinate
• Last generated X,Y values- inputs for next
  iteration

Probability Xn1 Yn1
0.01 0 0.16Yn
0.85 0.85Xn 0.04Yn -0.04Xn 0.85Yn 1.6
0.07 0.20Xn 0.26Yn 0.23Xn 0.24Yn 0.44
0.07 -0.15Xn 0.28Yn 0.26Xn 0.24Yn 0.44
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Computer-Generated Fractal Tree (100,000
iterations)
Computer-Generated Fractal Fern (100,000
iterations)
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Fractal Fern and Tree Data
Fern Dimension 1.5142
Tree Dimension 1.6222
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• Physics Fractals

• Two-Dimensional
• Gingerbread man map
• Lozi structure
• Henon structure
• Henon and Lozi structures used in
  calculating comet orbits
• Gingerbread man map derived from fluid
  equation
• Generated using recursive equations

• Three-Dimensional
• Rössler attractor
• Lorenz attractor
• Derived from Navier- Stokes equations
• Generated using differential equations

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Henon Structure 50,000 iterations
Lozi Structure 100,000 iterations
Gingerbread Man Map 100,000 iterations
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Rössler attractor 100,000 iterations
Lorenz attractor 100,000 iterations
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The Mandelbrot Set
Benoit Mandelbrot - 1975
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Mandelbrot Set

• z0 c
• zn1 zn2 c
• Points in set zn stays finite as n grows
  infinitely
• Coloring based on how quickly zn diverges

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Mandelbrot Set

• Self-Similarity

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Julia Sets

• Fix c in zn1 zn2 c
• Allow z0 to vary
• Each Julia set corresponds to a point in the
  Mandelbrot set

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Julia Sets

• Inside

• Outside

• Border

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Chaos Theory

• Developed through work of Edward Lorenz in
  1960s
• Led to famous butterfly effect
• Describes underlying order of random events
• Future behavior difficult or impossible to
  predict

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Bifurcation Graphs
Logistic equation- xn1rxn(1-xn)

• Single line starting point
• Branching
• Becomes dense and indecipherable
• Dimension 1.724

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Feigenbaums Constant

• Feigenbaum - 1975
• Describes functions approaching chaos
• Branches break off at certain decreasing values
  of r
• Limit as n approaches infinity of Ln/Ln1 where L
  is the length of a branch

• Approaches 4.669

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Bouncing Ball Simulation

• Bouncing Ball on a Vibrating Bed
• Ybed sine function
• Vibrational strength of bed described by G

G (A?2) / g
A amplitude of vibration ? angular velocity
of vibration g acceleration due to gravity
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Fractal Nature of the Simulation

• At small G values ball bounces with single
  definite collision frequency

• At larger G values ball stabilizes to bouncing
  with multiple frequencies

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Further Analysis of the Simulation

• Can frequency bifurcation be shown in a graph?
• Higher-precision program created
• Fourier transform to resolve frequencies
• No conclusive results
• Fourier transform insufficient
• More sophisticated analysis needed to get
  bifurcation
• Balls path multiple parabolas
• Possible properties
• Overall equation of path cycloid, complicated
• Collision frequency shows bifurcation, more
  numeric analysis needed

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Resonance

• For some G, maximum height greater than normal
• Collisions at same phase shift
• Ball receives same impulse

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Future Applications and Studies

• Analyze and Generate the Organ and Organelle
  Fractals
• All have fractal dimensions between 2 and 3--
  must be generated in 3 dimensions

Examples Include Brain Bronchial
Tubes Arteries Membranes

• Analyze and Generate Fractals in Additional
  Spatial Dimensions
• Analysis only through math and computers
• Box-Counting method not applicable