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Fractals

Fractals. Benoît Mandelbrot, “father of fractal geometry”. Jennifer Trinh. They’re SO BADASS!. I’m badass too!. Basic Idea. Fractals are Self-similar (will go into details in a moment) Cannot be described accurately with Euclidean geometry (they’re complex)

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Fractals

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  1. Fractals Benoît Mandelbrot, “father of fractal geometry” Jennifer Trinh

  2. They’re SO BADASS! I’m badass too!

  3. Basic Idea • Fractals are • Self-similar (will go into details in a moment) • Cannot be described accurately with Euclidean geometry (they’re complex) • Have a higher Hausdorff-Besicovitch dimension than topological dimension (will go into details in a moment) • Have infinite length or detail Romanesco Broccoli

  4. With Euclidean geometry…

  5. Exact Self-Similarity: Koch Snowflake Can be formed with L-systems

  6. Approximate Self-Similarity: Mandelbrot Set

  7. Statistical Self-Similarity

  8. Hausdorff-Besicovitch Dimension: Fractal Dimension? • relationship between the measured length and the ruler length is not linear, i.e.: 1 dimensional • The fractal/Hausdorff-Besicovitch dimension is d in the equation N = M^d, where N is the number of pieces left after an object is divided M times.  E.g., we divide the sides of a square into thirds, we have 9 total pieces left.  9 = 3^2, so the fractal dimension is 2. • More formally seen as log(N(l)) = log(c) - D log(l) • Doesn’t have to be an integer Sierpinski Triangle

  9. Generating Fractals • Escape-time fractals: give each point a value and plug into a recursive function (Mandelbrot set consists of complex numbers such that x(n+1)=x(n)^2 + c does not go to infinity, like i; they remain bounded). Depending on what a value does, that point gets a certain color, causes fractal picture • Iterated function systems: fixed geometric replacement • Random fractals: determined by stochastic processes (place a seed somewhere. Allow a particle to randomly travel until it hits the seed, then start a new randomly placed particle; see here) • “Escape-time fractals:

  10. “Measuring” Fractals • Smaller and smaller rulers • Box methods: counting the number of non-overlapping boxes or cubes (went over in Kenkel) • See Kenkel • Lacunarity: measuring how much space a fractal takes up (kind of like density). Another way to classify

  11. Sources • http://tiger.towson.edu/~gstiff1/fractalpage.htm • http://www.fractal-animation.net/ufvp.html • http://local.wasp.uwa.edu.au/~pbourke/fractals/ • http://www.fractalus.com/info/layman.htm • http://en.wikipedia.org/wiki/Fractal • http://mathworld.wolfram.com/KochSnowflake.html

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