The PascGalois Triangle: A Tool For Visualizing Abstract Algebra
The PascGalois Triangle: A Tool For Visualizing Abstract Algebra
Mike Bardzell and Kathleen M. Shannon
Salisbury State University
Abstract of article: In this paper we introduce a group-theoretical generalization of Pascal's triangle. We provide triangles constructed by choosing two (possibly the same) group elements, placing one down each side of the triangle and letting group multiplication generate the interior. The triangles presented here have a self-similar quality analogous to that of a fractal. Students can use these structures to visualize certain concepts from abstract algebra including subgroups, quotient groups, and automorphisms. Several student exercises are included for instructors to use in their abstract algebra courses.
About the authors: Michael Bardzell is an Assistant Professor of Mathematics in the Department of Mathematics and Computer Science at Salisbury University. He received a B.S. in Physics from Mary Washington College in Fredericksburg, Virginia and his M.S. and Ph.D. in Mathematics from Virginia Polytechnic Institute and State University. His research interests are ring theory, symbolic computation, and cellular automata. He regularly attends the National Conference on Undergraduate Research. Several of his students have already presented work related to the PascGalois project.
Kathleen Shannon is a Professor of Mathematics and Chair of the Department of Mathematics and Computer Science at Salisbury University. She received her B.A. in Mathematics and Physics from the College of the Holy Cross and her M.Sc. and Ph.D. in Applied Mathematics from Brown University. She is a student chapter coordinator for the Mathematical Association of America's Maryland/D.C./Virginia Section. Her interests include humanistic mathematics, scientific computing, approximation theory, general education, and mathematics education.
Software used: PascalGT
Color images to view: The Notes volume could only be printed in black-and-white. However, many of the graphics in this article are best viewed in color. (To do so, download the files given below and place in the same folder.)
Other links:
a web page intended to support this article
homepage for the PascGalois project
a Mathematica implementation for creating PascGalois triangles (based on AbstractAlgebra)
a Maple implementation for creating PascGalois triangles