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

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Presentation transcript:

HONR 300/CMSC 491 Fractals (Flake, Ch. 5) Prof. Marie desJardins, February 14, 2011 Fractals 2/14/11

Happy Valentine’s Day! Fractals 2/14/11

Key Ideas Self-similarity Fractal constructions Fractal widths/lengths Cantor set Koch curve Peano curve Fractal widths/lengths Recurrence relations Closed-form solutions Fractal dimensions Fractals in nature Fractals 2/14/11

Hilbert Curve Another space-filling curve Images: mathworld.com(T,L), donrelyea.com(R) Fractals 2/14/11

Koch Snowflake Same as the Koch curve but starts with an equilateral triangle Images: ccs.neu.edu(L), commons.wikimedia.org(R) Fractals 2/14/11

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) Fractals 2/14/11

Mandelbrot and Julia Sets ...about which, more soon!! Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org Fractals 2/14/11