MAT 103 ("Symmetries") : Course description, goals, bibliography*

"Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect. I hope I have not completely failed in giving you an indication of its many ramifications, and in leading you up the ladder from intuitive concepts to abstract ideas."

This concluding statement in eminent Princeton mathematician Herman Weyl's retiring address [7] remains as true today as it was many centuries ago: symmetry is an abstract concept celebrated in concrete ways through the art and culture of most known civilizations; it is this marriage between Life and Mathematics that led an archeologist and a mathematician to join forces and write a book [6] about it, used as a textbook in MAT 103 through Fall 97.

Students in MAT 103 learn to classify symmetric designs that either go around Roman mosaics, Christian icons, African baskets, Maori boats, etc ("border patterns") or completely fill mosque domes, Peruvian textiles, Chinese windows, Escher paintings [4], etc ("wallpaper patterns"); abstract symmetric designs are also found in purely mathematical works such as [2] and can be created by software packages like [1] or the instructors and students themselves. The classification of these patterns is based on the listing of the isometries (i.e., geometric transformations preserving distances therefore figures as well) that leave the pattern unchanged, and the investigation of the interactions among them.

Listing the isometries of a pattern simply amounts to the successful handling of a form of graphical data and what we call "critical seeing". To achieve that, students first go through a hands-on study of the effect of specific isometries (such as rotation and glide reflection) on specific figures, and they also learn to reconstruct the isometry from its effect. Investigating the interactions among such isometries requires a deeper geometrical insight (but still not extending much further than ordinary high school geometry) and leads naturally to the abstract, yet fundamental in nature, concept of a group. Students construct at least one group table arising from compositions of isometries, best understood in the context of their effect on color and appropriately colored tilings. Such tile or "map" colorings and their interplay with symmetry [3] can almost at the same time entertain the child and challenge the mathematician.

MAT 103 is a rather unconventional course, and this is best underlined by the absence of genuine textbooks in the area. Brilliantly written by an architect, [5] is probably the only book potentially suitable for the General Education student, but it fails to take color into consideration. We largely resolve this problem using handouts and encouraging team work during labs completed in class. We hope to facilitate the students' learning by the creation of a web page or, possibly, the publication of a locally produced textbook. After six years of successful implementation, MAT 103 is turning more and more mathematical. It is more technical, for example, than a similar course taught at Dartmouth and presented there during the Mathematics and Art Together in the Classroom workshop (June 26-28, 1997). Still, the students learn quite a bit about art and native cultures.

MAT 103's spirit is probably best captured in the words of a Gulf War veteran and Fort Drum soldier who graduated from SUNY Oswego in May 1997:

"I will never look at my Persian rugs the same way again."

References

[1] Edwards, Lois, and Kevin Lee, "TesselMania! Math Connection". Key Curriculum Press, 1995.

[2] Grunbaum, Branko, and G. C. Shephard, "Tilings and Patterns". W. H. Freeman and Co., 1987.

[3] Loeb, Arthur L., "Color and Symmetry". Wiley, 1971.

[4] Schattschneider, Doris, "Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher". W. H. Freeman and Co., 1990.

[5] Stevens, Peter S., "Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions". MIT Press, 1980.

[6] Washburn, Dorothy K. and Donald W. Crowe, "Symmetries of Culture: Theory and Practice of Plane Pattern Analysis". University of Washington Press, 1988.

[7] Weyl, Herman, "Symmetry". Princeton University Press, 1952.


*Statement prepared in April 1998 (when MAT 103 changed from "Symmetry and Culture" into "Symmetries" and was incorporated into the new General Education program of SUNY Oswego)