The Mandelbrot Set
A fractal defined by the complex quadratic polynomial z_{n+1} = z_n² + c, where z_0 = 0. Points that remain bounded under iteration form the set.
Formula: z_{n+1} = z_n² + c
Self-similarity appears at every scale when zooming into the boundary.
Julia Sets
For each complex constant c, the Julia set J_c consists of points z_0 whose orbits remain bounded under z_{n+1} = z_n² + c.
Connectedness: The Julia set is connected if c is in the Mandelbrot set, otherwise it's a Cantor set.
Barnsley Fern
An iterated function system (IFS) that generates a fern-like fractal. Each point is transformed by one of four affine transformations chosen probabilistically.
Transformations:
x_{n+1} = 0, y_{n+1} = 0.16y_n
x_{n+1} = 0.85x_n + 0.04y_n, y_{n+1} = -0.04x_n + 0.85y_n + 1.6
x_{n+1} = 0.20x_n - 0.26y_n, y_{n+1} = 0.23x_n + 0.22y_n + 1.6
x_{n+1} = -0.15x_n + 0.28y_n, y_{n+1} = 0.26x_n + 0.24y_n + 0.44
Fractal Tree (Pythagoras Tree)
A fractal constructed by recursively adding smaller branches to each limb, demonstrating self-similarity and exponential growth.
Number of branches: N = 2^{depth+1} - 2
Each branch is a scaled copy of the entire tree.
Box-Counting Dimension
A method to estimate the fractal dimension by covering the set with boxes of size ε and counting how many boxes N(ε) contain part of the fractal.
Results:
D = lim_{ε→0} (log N(ε) / log(1/ε))
The slope of log(N) vs log(1/ε) gives the fractal dimension.