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

What is a Fractal? Exhibits self-similarity Unique dimensionality Based on recursive algorithms Scale independent

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

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

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

Geometric Formula Smaller magnification improves accuracy Dimension = ln (self similar pieces) ln (magnification) D = ln 4 ln 2 2 4 Dimension Self-similar Squares Magnification Smaller magnification improves accuracy

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

Sierpinski Triangle

Sierpinski Triangle

Sierpinski “Square”

Sierpinski Carpet

Sierpinski Carpet

Other Sierpinski Polygons

The SURACE 17-gon

Sierpinski Chaos Game Vertex 1 Midpoint New Starting Point

Sierpinski Chaos Game 100 pts

Sierpinski Chaos Game 1000 pts

Sierpinski Chaos Game 5000 pts

Sierpinski Chaos Game 20000 pts

Sierpinski Triangle Data Fractal dimension = 1.8175…

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 Xn+1 Yn+1 0.01 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 -0.15Xn + 0.28Yn 0.26Xn + 0.24Yn + 0.44

Computer-Generated Fractal Tree (100,000 iterations) Computer-Generated Fractal Fern (100,000 iterations)

Fractal Fern and Tree Data Fern Dimension = 1.5142 Tree Dimension = 1.6222

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

Lozi Structure: 100,000 iterations Henon Structure: 50,000 iterations Gingerbread Man Map: 100,000 iterations

Rössler attractor: 100,000 iterations Lorenz attractor: 100,000 iterations

The Mandelbrot Set Benoit Mandelbrot - 1975

Mandelbrot Set z0 = c zn+1 = zn2 + c Points in set – zn stays finite as n grows infinitely Coloring based on how quickly zn diverges

Mandelbrot Set Self-Similarity

Each Julia set corresponds to a point in the Mandelbrot set Julia Sets Fix c in zn+1 = zn2 + c Allow z0 to vary Each Julia set corresponds to a point in the Mandelbrot set

Julia Sets Inside Outside Border

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

Bifurcation Graphs Logistic equation- xn+1=rxn(1-xn) Single line starting point Branching Becomes dense and indecipherable Dimension = 1.724

Feigenbaum’s Constant Describes functions approaching chaos Branches break off at certain decreasing values of r Limit as n approaches infinity of Ln/Ln+1 where L is the length of a branch Approaches 4.669

Bouncing Ball Simulation Bouncing Ball on a Vibrating Bed Ybed – sine function Vibrational strength of bed described by Γ Γ = (Aω2) / g A = amplitude of vibration ω = angular velocity of vibration g = acceleration due to gravity

Fractal Nature of the Simulation At small Γ values: ball bounces with single definite collision frequency At larger Γ values: ball stabilizes to bouncing with multiple frequencies

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 Ball’s path multiple parabolas Possible properties Overall equation of path – cycloid, complicated Collision frequency – shows bifurcation, more numeric analysis needed

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

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