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 geometrical structures that we mentally impose upon reality to make sense of it -- e.g., triangles, circles, ellipses, or tetrahedra. These patterns may also be found in structures that we create for any number of reasons. 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.

We are addressing primarily the *non-specialist* in mathematics, although we also include asides for mathematicians, particularly those who teach non-specialists. These asides can be found by clicking on the buttons with musical notes -- see the figure at the right, which in fact links to the first such aside. Pages intended for the more specialized audience will be clearly marked with these notes at the top of the page. Elsewhere we assume minimal background in mathematics, so the main body of the article 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, pages 8-11 will be difficult for some students to understand and would be more appropriate for mathematics majors in foundational courses, particularly those courses that introduce groups. If you find pages 8-11 too difficult, skim through them, skip to the patterns, or skip them altogether.

We hope this article may give you 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 being at some distance from mathematics.

### Acknowledgements

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 and Technology 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.

We thank undergraduate research student Andrew Nagel for providing the applets to accompany this article.

##### Published November, 2003

##### © 2003 by Kathleen M. Shannon and Michael J. Bardzell