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Mathematical Research The Mandelbrot Set

Mathematical Research The Mandelbrot Set. Amy K. Hoover COT 4810 Chapter 9: Mathematical Research January 29 th , 2008. [Wikipedia/Mandelbrot_set]. Outline. Mathematical Basics Fractals Complex numbers Connected sets vs totally disconnected The Mandelbrot Set Coding the Mandelbrot Set

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Mathematical Research The Mandelbrot Set

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  1. Mathematical Research The Mandelbrot Set Amy K. Hoover COT 4810 Chapter 9: Mathematical Research January 29th, 2008 [Wikipedia/Mandelbrot_set]

  2. Outline • Mathematical Basics • Fractals • Complex numbers • Connected sets vs totally disconnected • The Mandelbrot Set • Coding the Mandelbrot Set • Popular culture and research applications

  3. Mathematical Basics: Fractals • “Geometry that [describes] many of the irregular and fragmented patterns of nature” [Miranda 2001] • Property: Self-Similarity • Example: Approximate Self-Similarity (Cauliflower) Left: [goodnessdirect.co.uk] Right: [jupiterimages.com]

  4. Mathematical Basics: Fractals • Property: Fractional Dimensionality • Example: Measuring the coastline [Chris (picasa)]

  5. Mathematical Basics: Fractals • Property: Fractional Dimensionality • Example: Measuring the coastline [Chris (picasa)]

  6. Mathematical Basics: Fractals • Property: Fractional Dimensionality • Example: Measuring the coastline [Chris (picasa)]

  7. Mathematical Basics: Complex Numbers • Real Numbers < Complex Numbers • Form: z = a + bi • a in real • bi in imaginary, complex part • Adding Complex Numbers • Component-wise: (5 + 3i) + (-2 – i) = (3 + 2i) • Graphing Complex Numbers

  8. Mathematical Basis: Graphing Complex Numbers Graphing: 5 + 3i

  9. Mathematical Basis: Graphing Complex Numbers Graphing: -2 - 1i

  10. Mathematical Basics: Connected Sets • Connected Sets • Draw a "line" or curve between any two points of the set without leaving the set • E.g.

  11. Mathematical Basics: Totally Disconnected Sets • Totally disconnected sets • A line or curve can’t be drawn between any of the points in the set • E.g. (Cantor Dust [Wikipedia/Cantor_dust)

  12. The Mandelbrot Set [javaworld.com]

  13. The Mandelbrot Set • Mandelbrot Set: collection of points on the complex plane • Properties: • Connected • Centered around 0 or 0 + 0i • Self-Similar • Fractal boundary (e.g. coast line)

  14. The Mandelbrot Set • Defined by the quadratic recurrence relation: zi+1 = zi2 +C • Real Axis (-3, 1), Imaginary Axis (-2,2)

  15. The Mandelbrot Set: Self-Similarity • The Mandelbrot Set Self-Similarity: Misiurewicz, Feigenbaum points • “Mini-Mandelbrots” [Hubbard] [Wikipedia/Mandelbrot_set]

  16. The Mandelbrot Set • Recall equation: zi+1 = zi2 + C • Example of unbounded C [Miranda 2001] C = 1 (“drives the orbit to infinity”) z0 = 0 z1 = 02 + 1 = 1 z2 = 12 + 1 = 2 z3 = 22 + 1 = 5 z4 = 52 + 1 = 26 z5 = 262 + 1 = 677 …

  17. Mandelbrot Set: Code It! • Look at page 60 in the book for more detail • General Steps • Pick a region of the Mandelbrot Set to display and a maximum number of iterations on any particular C • Generate some kind of grid values of C, where C is the complex constant in zi+1 = zi2 +C, within the given area • For each C and each iteration, check |z|. If |z|<2, repeat until the max number of generations are calculated. If z reaches the maximum number of iterations, assign the color of the value C as black. Otherwise, assign the color of the C pixel as something distinguishable from black

  18. Mandelbrot Set Applications • Cool looking backgrounds • Interesting tattoos • Class topics • Music topics (http://www.mytoons.com/animation/play/7843) • Computer-Generated Music • FractMus (http://www.geocities.com/siliconvalley/haven/4386/) • MusiNum (http://reglos.de/musinum/) • Stock Market Predictions/Behavior[Herper02]

  19. BuddhaBrot

  20. BuddhaBrot

  21. References • [Dew93] A.K. Dewdney. The New Turing Omnibus. W.H. Freeman and Compant,1993. • [DM06] V Dolotin and A Morozov. The Universal Mandelbrot Set. World Scientific Publishing Co. Pte. Ltd., 2006. • [FM02] Michael L. Frame and Benoit B. Mandelbrot. Fractals, Graphics, and Mathematics Education. The Mathematical Association of America, 2002. • [Her] Matthew Herper. Mandelbrot: A math maverick takes stock. Online.http://www.forbes.com/2002/04/02/0402mandelbrot.html • [Man77] Benoit B. Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman and Company, 1977. • [Mir01] Eduardo Reck Miranda. Composing Music with Computers. Focal Press, 2001.

  22. Questions • 1. Cite one example from this presentation of a Mandelbrot Set application • 2. What is the iterative formula to generate a Mandelbrot Set on the complex plane

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