Newton's Method is an algorithm for finding the roots of real-valued functions or complex-valued functions. The image on the poster shows the basins of attraction for the five roots of the complex-valued function f(z) = z5 - 1. A root is simply a solution to the equation z5 - 1 = 0. The only real-valued solution is z = 1 and the basin of attraction for this root is colored yellow. The four other roots are complex and their basins of attraction are distinctly colored.
In 1898, Sir Arthur Cayley, a British mathematician, tried to determine the basins of attraction for the cubic function f(z) = z3 - 1, but failed. His failure ultimately led to the modern day study of fractals.
For more information see: http://en.wikipedia.org/wiki/Newton_fractal.