Searching for Patterns in Pascal's Triangle

Searching for Patterns in Pascal's Triangle
With a Twist
by Kathleen M. Shannon and Michael J. Bardzell
With Applets by Andrew Nagel
Department of Mathematics and Computer Science
Salisbury University
Salisbury, MD 21801

Introduction
Mathematics is, at heart, a search for patterns and for a deep understanding of how and why they occur. It does not matter if the patterns are in naturally occurring phenomena - e.g. weather or population growth - or in the geometrical structures that we mentally impose upon reality to make sense of it - e.g. triangles, circles, ellipses, tetrahedra, and other shapes. These patterns may also be found in structures that we create for any number of reasons. In this web "article" we will introduce you to some structures we have been studying and some of the questions that arise from investigating the patterns we see in them. The audience we are addressing here is primarily that of the non-specialist in mathematics, although we will also include asides for mathematicians, particularly those who teach non-specialists. These asides can be found by looking for the buttons with musical notes (see below). Any pages that are written for the more specialized audience will be clearly marked with these notes at the top of the page. We have done our best to write the remaining pages assuming minimal background in mathematics so that they may be enjoyed by students, particularly those in Liberal Arts Mathematics or Mathematics Appreciation courses, or anyone who is interested in gaining a new appreciation for mathematics. However, it may be that section 2 will be difficult to understand for some students and would be more appropriate for mathematics majors in foundational courses, particularly those which introduce groups. Readers who find section 2 too difficult may skim through it, skip to the patterns or skip it altogether. We hope that this article may give the reader an appreciation for the beauty of mathematics and an understanding of those who study it for its own sake. We also hope to point out some connections to disciplines and areas of study which, unlike most of the sciences, are frequently seen as at some distance from mathematics.

Section 0: What is Pascal's Triangle

Section 1: What is the twist? and How about the Patterns?

Section 2: Another twist and More Patterns?

Section 3: What about the way humans perceive patterns?

Section 4: Some extensions beyond the triangle

Print Works Cited and Web References

Support for much of this work was provided by the National Science Foundation award # DUE-0087644
and by the Richard A. Henson endowment for the School of Science at Salisbury University.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
The authors would like to thank undergraduate research student Andrew Nagel for providing the applets to accompany this article.