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

HONR 300/CMSC 491 Fractals (Flake, Ch. 5). Prof. Marie desJardins, February 15, 2012. Happy Valentine’s Day!. Key Ideas. Self-similarity Fractal constructions Cantor set Koch curve Peano curve Fractal widths/lengths Recurrence relations Closed-form solutions Fractal dimensions

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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 15, 2012

  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. Cantor Sets • Construction and properties (activity!) • Description of points in Cantor set • Standard Cantor set: “middle third” removal • Variation: “middle half” • Distance between pairs of end points at iteration i = ? • Width of set at iteration i = ?

  5. Fractional dimensions • D = log N / log(1/a) • N is the length of the curve in units of size a • Cantor set: D = ? • Koch curve: D = ? • Peano curve: D = ? • Standard Cantor: D = ? • Middle-half Cantor: D = ?

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

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

  8. 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)

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

  10. Fractals in Nature • Coming up soon!!

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