ISOMETRICA INTRODUCTION -- AUGUST 2001


Professor Groman introduced MAT 103 ("Symmetry and Culture") in Fall 1992, as a course fulfilling both the Human Diversity and the Mathematics requirements of the General Education program at SUNY Oswego. In time MAT 103 became more mathematical, with two-colored wallpaper patterns introduced in Fall 1995, colorings of tilings in more than two colors in Fall 1996, compositions of isometries in the context of colored tilings in Spring 1997, and algebraic description of planar isometries (distance-preserving transformations of the plane) in Fall 1998. In view of the restructuring of SYNY Oswego's General Education program in the late 90's, it became clear that we could achieve only one of the course's original goals: MAT 103 was approved as a Mathematics course ("Symmetries") in the Knowledge & Foundations area of General Education in Spring 1998, and the writing of these notes began immediately afterwards.

MAT 103's evolution and renaming does not mean that we have completely abandoned the course's cultural component: chapter 2 (border patterns) and chapter 4 (wallpaper patterns) are spiced with symmetrical designs conceived all over our small world and during several historical periods; and I hope to soon create a web page presenting such material in color and in context, part of http://www.oswego.edu/~baloglou/103 (where various developments, additional material, exercises and examples will also be posted).

If you happen to be a professor, and a Mathematics professor in particular, reading this, please keep in mind that our work is mainly addressed to General Education students with a weak mathematical background and is meant to be read by them. This does not mean that there will be no challenges here for the Mathematics major or even instructor, especially after the addition of exercises: indeed I hope that the final version of this work will be appropriate as a textbook or supplement for a college or high school geometry course.

If you are a student, and a MAT 103 student in particular, please keep in mind that teaching symmetry at this level is essentially a novel enterprise. Although MAT 103 itself has been a very successful course so far, the success of this book-in-the-making depends a lot on your reaction and comments. I have made every effort to produce something readable and even pleasant, but it is inevitable that there are shortcomings to be discovered and improvements to be made: if ever in doubt when going over these notes, please let your instructor know! And if you are not that studious or mathematically inclined, keep in mind that these notes may be read at several levels: you can always start from the pictures, then move into those comments that elaborate on the pictures, then read everything else -- it works!

A few comments on the mathematical content of this work now. The main goal is an alternative, completely geometrical and elementary, classification of wallpaper patterns: this is done in chapter 9 [8], relying on techniques and ideas from chapter 8 [7], where isometries and their pattern-creating interactions are studied in some depth; some of these techniques and ideas arise naturally in chapter 6, where investigation of two-colored possibilities for wallpaper patterns is used as an incentive to informally delve into their structure. Coloring is introduced in chapter 5, and in the simpler context of two-colored border patterns, the investigation of which in chapter 2 serves as a prelude for the introduction to wallpaper patterns in chapter 4. Isometries themselves are studied in chapters 1 and 3, both of which may in fact be covered after, or in parallel with, chapter 2 or even chapter 4. Chapter 7 [9] is the only one that remains incomplete at present: it aims at an investigation of wallpaper pattern structure in the context of multi-colored tilings, and will probably be posted on the MAT 103 web site during the 2001-2002 academic year.

The overall approach is very geometrical and rather informal. An algebraic description of isometries is offered as a way of connecting MAT 103 to the broader mathematical world, but is not essential to the main goal. Groups and fundamental regions are certainly mentioned, but are not at all central in the presentation. Pattern classification flowcharts have been replaced by symmetry plans, which, in our opinion, allow for a better view and understanding of the patterns. Border patterns are viewed as building blocks of wallpaper patterns, and the process of passing from one pattern type to another is dynamic. Color is used to both appeal to the artistically inclined reader and reveal the group structure hidden behind the patterns to the mathematically minded.

I am indebted to Margaret Groman for her encouragement and many conversations and insights over the last six years, as well as collaboration during the first stages of the project. I have also received valuable suggestions from Mark Elmer, who used an earlier version of these notes during the Fall 2000 semester, and from my former colleague and collaborator Phil Tracy. Special thanks are due to Fred Barber, Jim Burling, Doug Deal, Phil Downum, Matthew Friday, Jack Narayan, Sue Weber and Helen Zakin for their assistance at crucial moments. And I would like to express my gratitude to a number of former students who helped me to improve MAT 103 in a variety of ways; in particular, Michael Nichols has provided valuable assistance with the graphics.