# Newton's Method Fractal for z3 - 1 = 0

Consider the roots for the equation z^3 -1 = 0. In the Real domain there is only a single solution: 1. Over the complex plane however there are two additional roots: -1/2 + sqrt(3)/2 i and -1/2 - sqrt(3)/2 i. For any starting complex number we can use Newton's method to successively approach one of the roots. Newton's method is:

x_1 = x_0 - f(x_0)/f^'(x_0) which in this case works out to:  x_1 = x_0 - (x_0^3 - 1)/(3x_0^2)

Which root we approach will vary depending on the initial complex number chosen and that's what this fractal represents. Each pixel represents a complex number to which we apply several iterations of Newton's method. If we converge to one of the roots we color it accordingly. The hue represents which of the roots was approximated and the brightness encodes how quickly that solution was approached; that is, the number of iterations to get within some ε of the true root.