The Mandelbrot set is the result of taking the function [math]f(z) = z^2 + c[/math] and iterating it with itself over and over an infinite number of times, where we start with zero.
[math]\displaystyle 0 \mapsto f(0) \mapsto f(f(0)) \mapsto f(f(f(0))) ...[/math]
This function may look simple, but it does a pretty good job at mixing things up when iterated.
We choose [math]c[/math] to be a value on the complex plane, and then we color that point based on how long it takes for the process to get too big, and another color (usually black) if it always stays small.
Some values are simple. If [math]c=0[/math] then [math]0[/math] stays [math]0[/math], if [math]c=3[/math] then it just blows up
The word “fractal” is only loosely defined, but everyone agrees that fractals share two general kinds of properties:
- Fractal dimension: A fractal has a fractal (Hausdorff) dimension exceeding its topological dimension. Roughly speaking, this means that a fractal is so infinitely “bumpy” that scaling it up causes its measure to increase faster than you would expect from a non-fractal.
- Self-similarity: Zooming in to a fractal yields pictures that are similar (whether exactly or somehow approximately) across widely varying scales.
The Mandelbrot set has fractal properties of both types.
Fractal dime
“Weirdness” of the Mandelbrot and Julia sets arises from a few simple mathematical facts that produce extreme complexity: iteration, nonlinearity, and sensitivity to initial conditions. The shapes are direct visualizations of how a simple quadratic map behaves under repeated application.
- Iteration of a complex quadratic map: z ↦ z² + c. Repeated application (z₀, z₁, z₂, …) generates dynamics for each complex parameter c and/or initial seed z₀.
- Dichotomy (escape vs. boundedness): Points are classified by whether their orbit remains bounded or escapes to infinity. The boundary betw