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1. What do these pictures have in common? Write a complete sentence.

1. What do these pictures have in common? Write a complete sentence. What are Fractals?. A fractal is a rough or fragmented geometric shape can be broken into parts each part is a smaller copy of the whole. Self Similarity.

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1. What do these pictures have in common? Write a complete sentence.

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  1. 1. What do these pictures have in common? Write a complete sentence.

  2. What are Fractals? • A fractal is a rough or fragmentedgeometric shape • can be broken into parts • each part is a smaller copy of the whole.

  3. Self Similarity • If a fractal’s parts are copies of the whole the fractal is self similar. • If any given part of the fractal is an exact replica of the whole fractal, the fractal is strictly self similar. • A fractal is entirely self similar or strictly self similar, it cannot be part one thing and part another. • Go to the website below, then return to this PowerPoint. http://astronomy.swin.edu.au/~pbourke/fractals/selfsimilar/

  4. Traits of Fractals • Self-similarity-smaller regions resemble the entire diagram when we zoom in on specific areas • “Strictly” self-similar fractals- made of exact copies of the original put together. • “Recursively” self-similar - have some of the same shapes in smaller sections as in the bigger sections.

  5. Base shape Basic change 1st iteration 2nd iteration To make a fractal • Begin with a base shape and a basic change • Make the change to each successive stage of the fractal. Each new stage of the fractal is called an iteration.

  6. Base shape Basic change 1st iteration 2nd iteration Another example

  7. MORE EXAMPLES • For the next few examples, continue to press enter so the fractal is generated. You will see the following. • Tree • Sierpinski’s Triangle • Dragon Curve • Koch’s Snowflake

  8. Tree

  9. Stage one Stage five Stage three Stage seven Stage six Stage two Stage four

  10. Dragon Curve

  11. Koch Snowflake

  12. PART II: Fractal dimension What is dimension? How do we assign dimension to an object? • When you see: • a train moving along railroad tracks, 1. In what dimension does it move?

  13. PART II: Fractal dimension What is dimension? How do we assign dimension to an object? • When you see: • a boat sailing on a lake, 2. In what dimension does it move?

  14. PART II: Fractal dimension What is dimension? How do we assign dimension to an object? • When you see: • a plane in the sky, 3. In what dimension does it move?

  15. Fractal dimension • I. Take an unused piece of aluminum foil: • What is it's dimension? • II. Now, crumple it up into a ball: • What is the dimension of the ball of foil? • III. When you carefully reopen the ball of foil, what dimension has it become?

  16. PART II: Fractal dimension 4. What is the dimension of a fractal between? ________________ Fractal dimension reference http://www.math.umass.edu/~mconnors/fractal/dimension/dim.html Go to the above link for more information about fractal dimension.

  17. PART III. Iteration and orbits • Use the worksheet and the following link at the same time. http://aleph0.clarku.edu/~djoyce/julia/julia.html For more details, see http://www.jcu.edu/math/vignettes/population.htm http://www.ies.co.jp/math/java/misc/chaosa/chaosa.html

  18. PART IV: Major Types of Fractals Julia Mandelbrot Sierpinski Triangle Koch Snowflake

  19. The Mandelbrot Set • Probably the most well known fractal is the Mandelbrot Set. • The Mandelbrot Set is a group of complex points that have a magnitude limit of 2 when iterated in zn+1= zn2 + c • The Map is the graph of the points tested (the points in the black area are within the Mandelbrot set while the colored points are not)

  20. The Mandelbrot Graph • Coloring of the points tested for a Mandelbrot set varies • Arbitrary color assignments based on number of iterations it takes for the magnitude of a point to become larger than 2 are used.

  21. PART IV: The Mandelbrot Set Use the below website to complete PART IV of your worksheet along with your calculator. http://www.geocities.com/CapeCanaveral/2854/

  22. Gaston Julia • Gaston Julia was one of the first to work with the limits of Fractals. His question was based on the bounds of fractals with a given C. He asked for what values of Z does the equation stay bounded. So to find a number that left the equation bounded he fixed a value to C and so created the instructions for making a Julia set of numbers. First fix a value to C and then find all Zs that leave Z2 + C bounded.

  23. The Julia Set f(z) = z2 + c

  24. The Julia Set f(z) = z2 + c Go to the following website and read the complex number example. http://aleph0.clarku.edu/~djoyce/julia/julia.html If you want more information, you can read more about Julia Sets here http://www.geocities.com/CapeCanaveral/2854/ and click on Julia Sets on the left hand side. http://www.mcgoodwin.net/julia/juliajewels.html

  25. Go to the following link and explore the sets using the applets. http://nlvm.usu.edu/en/nav/frames_asid_136_g_3_t_3.html?open=instructions We hope you enjoyed learning about Fractals. Next, your group will explore a particular fractal and teach your classmates.

  26. References • Some of the slides were part of previous Honors Precalculus Classes at Hinsdale South High School. In addition, several websites have been used to help you understand the concepts.

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