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

They’re SO BADASS! I’m badass too!

Basic IdeaBasic 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

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 picturefractal picture Iterated function systems: fixed geometric replacementfixed 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)here

“Measuring” Fractals Smaller and smaller rulerssmaller 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 mlhttp://mathworld.wolfram.com/KochSnowflake.ht ml