Description
Speaker: Kathleen Shannon, Salisbury University.
Title: Pascal's Triangle, Cellular Automata and Serendipity: A Mathematical Tale
Abstract: The talk will outline the development of the PascGalois Project. Its origins are in an exercise using Pascal's Triangle and modular arithmetic. Colors are assigned to the numbers 0, 1, ..., n-1, and Pascal's Triangle modulo n is drawn. The patterns in the triangle are then related to the properties of the cyclic group Zn. The process of drawing the triangles is then generalized to non-cyclic and non-abelian groups and the new patterns are examined in light of the properties of these groups. The images can help develop visual and intuitive understanding of concepts such as subgroup closure and quotient groups. They can also be used to discuss the relationship between mathematical properties and visual aesthetics. Finally we view Pascal's Triangle as a one-dimensional cellular automata and generalize to more general initial conditions and two dimensional automata. Many of the investigations in this project have been undertaken with students in undergraduate research projects and one outgrowth of the project has been the development of a set of visualization exercises to supplement the standard undergraduate course in abstract algebra. The web site for the project is at www.pascgalois.org.