HONR 300/CMSC 491 Fractals (Flake, Ch. 5) Prof. Marie desJardins, February 15, 2012 Fractals 2/15/12

Happy Valentine’s Day! Fractals 2/15/12

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 2/15/12

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 = ? Fractals 2/15/12

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 = ? Fractals 2/15/12

Hilbert Curve Another space-filling curve Images: mathworld.com(T,L), donrelyea.com(R) Fractals 2/15/12

Koch Snowflake Same as the Koch curve but starts with an equilateral triangle Images: ccs.neu.edu(L), commons.wikimedia.org(R) Fractals 2/15/12

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 2/15/12

Mandelbrot and Julia Sets ...about which, more soon!! Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org Fractals 2/15/12

Fractals in Nature Coming up soon!! Fractals 2/15/12