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HONR 300/CMSC 491 Fractals (Flake, Ch. 5). Prof. Marie desJardins, February 14, 2011. 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 491Fractals (Flake, Ch. 5) Prof. Marie desJardins, February 14, 2011
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
Hilbert Curve • Another space-filling curve Images: mathworld.com(T,L), donrelyea.com(R)
Koch Snowflake • Same as the Koch curve but starts with an equilateral triangle Images: ccs.neu.edu(L), commons.wikimedia.org(R)
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)
Mandelbrot and Julia Sets • ...about which,more soon!! Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org