I got better!
Basins of attraction for Newton’s method on \(z^3-1=0\).
Basins of attraction for Newton’s method on \(z^5-1=0\).
Newton’s Method is an iterative numerical technique for finding the roots, or zeroes, of a function. You pick an arbitrary point, and perform a procedure on that point using the function of interest, getting another, new point, which you perform the same procedure on, getting a new point, and so on.
This procedure is constructed so that its fixed points are the zeroes of the function you’re insterested in, and is guaranteed to yield an approximation to one of the zeroes… given enough time.
We can naturally extend this procedure into the complex plane, where we then can color each point based on which zero it approaches, and how long it takes for this procedure to “settle down”. When we make this coloring we get objects known as Newton’s Fractal.