1. 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

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

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

4. 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

5. 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

6. 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

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

8. Sierpinski Triangle

9. Sierpinski Triangle

10. Sierpinski “Square”

11. Sierpinski Carpet

12. Sierpinski Carpet

13. Other Sierpinski Polygons

14. The SURACE 17-gon

15. Sierpinski Chaos Game Vertex 1 Midpoint New Starting Point

16. Sierpinski Chaos Game
100 pts

17. Sierpinski Chaos Game
1000 pts

18. Sierpinski Chaos Game
5000 pts

19. Sierpinski Chaos Game
20000 pts

20. Sierpinski Triangle Data
Fractal dimension = …

21. 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 Yn
-0.04Xn Yn + 1.6
0.07
0.20Xn – 0.26Yn
0.23Xn Yn
-0.15Xn Yn
0.26Xn Yn

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

23. Fractal Fern and Tree Data
Fern Dimension =
Tree Dimension =

24. 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

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

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

27. The Mandelbrot Set
Benoit Mandelbrot

28. 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

29. Mandelbrot Set
Self-Similarity

30. 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

31. Julia Sets
Inside
Outside
Border

32. 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

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

34. 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

35. 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

36. 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

37. 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

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

39. 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