The pseudo-Anosov homeomorphism is one of the most important concepts in the areas of mathematics known as three-dimensional geometry, three-dimensional topology, and two-dimensional dynamical systems. The study of pseudo-Anosov homeomorphisms began in the mid-1970's. During the decade that followed, a great deal of progress was made in analyzing their behavior. Since then progress has slowed, in part because only the more difficult problems remain. Chief among these problems is an understanding of the "forcing partial order," which, roughly speaking, describes the way in which simple two dimensional systems change into chaotic ones. I use extensive computer calculations and techniques from other areas of mathematics, especially that of geometric group theory to investigate this problem.