Among the methods available for the characterization of complicated mathematical and physical phenomena, computers with graphics are emerging as an important tool. In this article, I present computational and graphical results on Halley's method for one-parameter functions of the form ζ(ζ^α − 1) = 0 and sin(ζ) = 0 in order to gain insight as to where the method can be relied upon and where it behaves strangely. The resulting plots reveal a visually striking and intricate class of patterns indicating behavior ranging from stable attractive and repulsive points to chaos. Iterative approximation methods such a Halley's method occur frequently in science and engineering.