@@ -27,14 +27,24 @@ The **Mandelbrot set** is the set of **complex numbers** $c$ for which the funct
> * For each point $c$, we iterate and check whether $\left\lvert z_{n} \right\rvert$ (the magnitude of $z_{n}$) stays bounded (i.e., does not grow beyond a threshold like $2$).
> * If the value of $\left\lvert z_{n} \right\rvert$ grows beyond $2$, then the point $c$ **is not** part of the Mandelbrot set (i.e., it escapes to infinity).
### 5. Check if $\left\lvert z \right\rvert$ (the modulus of $z$) escapes
### 5. Check if $\left\lvert z_{n} \right\rvert$ (the modulus of $z_{n}$) escapes
> * **Escape condition**: If the magnitude $\left\lvert z_{n} \right\rvert$ of the current value of $z$ becomes greater than $2$, we stop iterating because the value will continue growing towards infinity. This means that the point $c$ is **not part of the Mandelbrot set**.
> * If $\left\lvert z_{n} \right\rvert$ stays below $2$ after a certain number of iterations, then the point $c$ is considered **part of the Mandelbrot set**.
### 6. Repeat the iteration for a maximum number of iterations
> * The number of iterations we run is often limited to a maximum (like $100$). If the value of $z$ does not escape within that many iterations, we consider it part of the Mandelbrot set.
### 7. Color the point
> * Points that escape quickly (i.e., after just a few iterations) can be colored differently from those that take longer to escape.
> * Points that do **not escape** (after the maximum number of iterations) are part of the Mandelbrot set and are typically colored black.
### 8. Repeat for all points in the complex plane
> * After performing these iterations for each point in the grid of complex numbers, you can plot the results.
