1. David Chan TCM 2004 --and what can you do with it in class?

2. Outline What are Fractals? -Build a Fractal Dimension -Measure the Fractal Dimension of different objects How are Fractals constructed? -Basic Fractals and their properties -L-systems and Function Composition/Iteration -Derivatives and the Complex Plane Summary

3. What is a Fractal? A rough, fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole.—Benoit Mandelbrot (Mathematical) A set of points whose fractal dimension exceeds its topological dimension. “An object whose dimension is not an integer.”

4. Examples

5. Can we construct one? Fractal Dimension Hint: Because Fractals have a self-similarity Property, we can use boxes to measure their Dimension. Hint(2): Look at a ratio of number of boxes to the size of the boxes. Hint(last): Look at the ratio of some function of the number of boxes to the size of the boxes

6. Fractal Dimension? Try some basic objects. Try some fractal objects! Does it make sense? Oh well, try again. Due to time constraints the answer is…

8. Dimension (cont.) Box dimension is calculated using: where N(d,F) is the smallest number of sets of diameter d which can cover F.

9. How are fractals constructed? Geometrical Process Function Composition Function Attractors

10. Koch Snowflake

11. Sierpinski’s Triangle

12. Cantor’s Middle Thirds Set

13. L-systems Example: Start off with a rule F  FF(LF)(RF) And an initial string F Then compose/iterate

14. F FF(RF)(LF) FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF)) FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF)) FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF)) (L FF(RF)(LF))(R FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF)))(L FF(RF)(LF)FF(RF)(LF) (R FF(RF)(LF))(L FF(RF)(LF)) FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF)) (LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF) (RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(RFF(RF) (LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF) (LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF) (LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))))

17. Attractors When a function, say, is iterated starting with some value, say, then an orbit is created. This orbit, or sequence, is written as

18. Under certain conditions, orbit can converge (or limit on) a particular set of point(s). These sets are called attractors. Types of attractors: Fixed points Periodic orbits Strange attractors

19. Chaos Game http://www.shodor.org/interactive/activities/chaosgame An Example of systems that give attractors:

20. Examples of keeping track of attractors Julia Sets Mandelbrot Sets

21. -Everyone’s favorite curved function: -Complex Plane -Complex Arithmetic -Graphing Complex Functions -Complex DERIVATIVES!

22. COMPLEX DERIVATIVES! Definition: For a complex function F(z), we define it’s complex derivative, F’(z), to be

25. Summary Algebra/Geometry-Look at fractals and do simple calculations. Play with the Chaos game. Precalculus-Shifting/Stretching pictures, L-systems and composition, and do some numerical experiments. Calculus-Talk about attractors and complex differentiation. Beyond Calculus-Proofs, write programs to create fractals.