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

Prof. Marie desJardins, February 15, 2012

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Happy Valentine’s Happy Valentine’s Day!Day!

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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

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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 = ?

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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 = ?

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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!!

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