The Mandelbrot Set $M$ is the [[Set]] of all [[Point|Points]] within the [[Complex Plane]] $\C$, where the point $c \in \C$ remains [[Finite]] when recursively applying this [[Function]]: $\huge \begin{align} z_{0}(c) &= 0 \\\ z_{n+1}(c) &= z_{n}^2 + c \\ \\ \end{align}$ $ \huge c \in M \iff \lim_{n\to\infty} z_{n} \text{ remains finite} $ ![[../../00 Asset Bank/Pasted image 20241120142459.png|sepia]] >[!note] >In these images, only the points colored black are part of the set, all other points are mapped to some color dependent on how quickly they blow up to [[Infinity]]. ```rust const ITERATIONS: usize = 100; fn inside_set(c: Complex) -> bool { let mut z = Complex::new(0., 0.); for i in 0..ITERATIONS { z = z * z + c; } return z.abs() < 2.; } ```