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HONR 300/CMSC 491 Fractals (Flake, Ch. 5)

Discover the beauty of self-similarity through fractal constructions like Cantor and Koch curves. Explore fractal dimensions and their presence in nature while marveling at the intriguing Sierpinski Triangle. Dive into the world of Mandelbrot and Julia sets for a deeper understanding. Happy Valentine’s Day!

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HONR 300/CMSC 491 Fractals (Flake, Ch. 5)

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  1. HONR 300/CMSC 491Fractals (Flake, Ch. 5) Prof. Marie desJardins, February 14, 2011

  2. Happy Valentine’s Day!

  3. Key Ideas • Self-similarity • Fractal constructions • Cantor set • Koch curve • Peano curve • Fractal widths/lengths • Recurrence relations • Closed-form solutions • Fractal dimensions • Fractals in nature

  4. Hilbert Curve • Another space-filling curve Images: mathworld.com(T,L), donrelyea.com(R)

  5. Koch Snowflake • Same as the Koch curve but starts with an equilateral triangle Images: ccs.neu.edu(L), commons.wikimedia.org(R)

  6. Sierpinski Triangle • Generate by subdividing an equilateral triangle • Amazingly, you can also construct the Sierpinski triangle with the Chaos Game: • Mark the three vertices of an equilateral triangle • Mark a random point inside the triangle (p) • Pick one of the three vertices at random (v) • Mark the point halfway between p and v • Repeat until bored • This process can be used with any polygon to generate a similar fractal • http://www.shodor.org/interactivate/activities/TheChaosGame/ Images: curvebank.calstatela.edu(L), egge.net(R)

  7. Mandelbrot and Julia Sets • ...about which,more soon!! Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org

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