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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 - PowerPoint PPT Presentation
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HONR 300/CMSC 491Fractals (Flake, Ch. 5)
Prof. Marie desJardins, February 15, 2012
1 Fractals 2/15/12
Happy Valentine’s Happy Valentine’s Day!Day!
Fractals 2/15/122
Key IdeasKey Ideas Self-similarity Fractal constructions
Cantor set Koch curve Peano curve
Fractal widths/lengths Recurrence relations Closed-form solutions
Fractal dimensions Fractals in nature
Fractals 2/15/123
Cantor SetsCantor 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 2/15/124
Fractional dimensionsFractional 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 2/15/125
Hilbert CurveHilbert Curve Another space-filling curve
Fractals 2/15/126Images: mathworld.com(T,L), donrelyea.com(R)
Koch SnowflakeKoch Snowflake Same as the Koch curve but starts with an equilateral
triangle
Fractals 2/15/127Images: ccs.neu.edu(L), commons.wikimedia.org(R)
Sierpinski TriangleSierpinski 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/
Fractals 2/15/128Images: curvebank.calstatela.edu(L), egge.net(R)
Mandelbrot and Julia Mandelbrot and Julia SetsSets
...about which,more soon!!
Fractals 2/15/129Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org
Fractals in NatureFractals in Nature Coming up soon!!
Fractals 2/15/1210