FractalsJennifer Trinh

Benoît Mandelbrot, “father of fractal geometry”

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)

•Have a higher Hausdorff-Besicovitch dimension than topological dimension (will go into details in a moment)

•Have infinite length or detailRomanesco

Broccoli

With Euclidean geometry…

Exact Self-Similarity: Koch Snowflake

Can be formed with L-systems

Approximate Self-Similarity: Mandelbrot Set

Statistical Self-Similarity

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

Generating Fractals•“Escape-time 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)

“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

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