Motivated by the mesmerizing qualities of three animated points in a computer microworld, this thesis explores the average triangular area determined three particles on a random walk on . Intermediate steps include the study of distance between two walking points in one and two dimensions, and the average area of three random points on . The distance functions are shown to grow on the order , while area appears to grow linearly. Relationships between the distance results and area investigation are explored while some curious contradictions between an unbounded and bounded region arise in data collected through modeling the situations. Numerous questions for further study are presented.
