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The infinitely complex… Fractals

The infinitely complex… Fractals. Jennifer Chubb Dean’s Seminar November 14, 2006 Sides available at http://home.gwu.edu/~jchubb. Fractals are about all about infinity…. The way they look, The way they’re created, The way we study and measure them…

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The infinitely complex… Fractals

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  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. Fractal Gallery 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!

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