1. The infinitely complex… Fractals Jennifer Chubb Dean’s Seminar November 14, 2006 Sides available at http://home.gwu.edu/~jchubb

2. Fractals are about all about infinity… The way they look, The way they’re created, The way we study and measure them… underlying all of these are infinite processes.

3. Fractal Gallery 3-Dimensional Cantor Set

4. Fractal Gallery Koch Snowflake Animation

5. Fractal Gallery Sierpinski’s Carpet Menger Sponge

6. Fractal Gallery The Julia Set

7. Fractal Gallery The Mandelbrot Set

8. Dynamically Generated Fractals and Chaos Chaotic Pendulum http://www.myphysicslab.com/pendulum2.html

9. Fractal Gallery Henon Attractor http://bill.srnr.arizona.edu/classes/195b/henon.htm

10. Fractal Gallery Tinkerbell Attractor and basin of attraction

11. Fractal Gallery Lorenz Attractor

12. Fractal Gallery Rossler Attractor

13. Fractal Gallery Wada Basin

14. Fractal Gallery

15. Romanesco – a cross between broccoli and cauliflower

16. What is a fractal? Self similarity As we blow up parts of the picture, we see the same thing over and over again…

17. What is a fractal? So, here’s another example of infinite self similarity… and so on … But is this a fractal?

18. What is a fractal? No exact mathematical definition. Most agree a fractal is a geometric object that has most or all of the following properties…  Approximately self-similar  Fine structure on arbitrarily small scales  Not easily described in terms of familiar geometric language  Has a simple and recursive definition  Its fractal dimension exceeds its topological dimension

19. Dimension Topological Dimension Points (or disconnected collections of them) have topological dimension 0. Lines and curves have topological dimension 1. 2-D things (think filled in square) have topological dimension 2. 3-D things (a solid cube) have topological dimension 3.

20. Dimension Topological Dimension 0 The Cantor Set (3D version as well)

21. Dimension Topological Dimension 1 Koch Snowflake Chaotic Pendulum, Henon, and Tinkerbell attractors Boundary of Mandelbrot Set

22. Dimension Topological Dimension 2 Lorenz Attractor Rossler Attractor

23. Dimension What is fractal dimension? There are different kinds: Hausdorff dimension… how does the number of balls it takes to cover the fractal scale with the size of the balls? Box-counting dimension… how does the number of boxes it takes to cover the fractal scale with the size of the boxes? Information dimension… how does the average information needed to identify an occupied box scale? Correlation dimension… calculated from the number of points used to generate the picture, and the number of pairs of points within a distance ε of each other. This list is not exhaustive!

24. Box-counting dimension Computing the box-counting dimension… … … … … and so on… 1.26186

25. Hausdorff Dimension of some fractals… Cantor Set… 0.6309 Henon Map… 1.26 Koch Snowflake… 1.2619 2D Cantor Dust… 1.2619 Appolonian Gasket… 1.3057 Sierpinski Carpet… 1.8928 3D Cantor Dust… 1.8928 Boundary of Mandelbrot Set… 2 (!) Lorenz Attractor… 2.06 Menger Sponge… 2.7268

26. Thank you!