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Exploring Fractals: Self-Similarity & Nature's Beauty

Delve into the world of fractals, from self-similarity and construction methods to Cantor sets and nature's use of fractals. Discover the beauty of Koch curves, Peano curves, and more. Dive into the dimensions and patterns found in fractals.

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Exploring Fractals: Self-Similarity & Nature's Beauty

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

  2. Happy Valentine’s Day!

  3. 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

  4. 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 = ?

  5. 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 = ?

  6. Hilbert Curve • Another space-filling curve Images: mathworld.com(T,L), donrelyea.com(R)

  7. Koch Snowflake • Same as the Koch curve but starts with an equilateral triangle Images: ccs.neu.edu(L), commons.wikimedia.org(R)

  8. 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)

  9. Mandelbrot and Julia Sets • ...about which,more soon!! Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org

  10. Fractals in Nature • Coming up soon!!

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