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Bibliography

[Ce] Cederberg, Judith, A Course in Modern Geometries, Springer-Verlag, 1989.

An elegant textbook for an introductory course in geometry, this presents the material on friezes in detail, leaving wallpaper for the exercises.

[FS] Farlow, Stanley J., Partial Differential Equations for Scientists and Engineers, Dover Publications, Inc, NY, 1993
[Fa] Farris, Frank A, "Review of Visual Complex Analysis by Tristan Needham", Amer. Math. Monthly 105 (1998), to appear in June-July issue.
[Fi] Field, Michael and Martin Golubitsky, Symmetry in chaos: a search for pattern in mathematics, art and nature, Oxford University Press, Oxford, 1992.
[Ga] Garabedian, P. R., Partial Differential Equations, Chelsea Publishing, NY, 1986.
[Gr] Gruenbaum, Branko and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1986.

This is considered the bible of the subject, summarizing most of what is known about the subject. The approach is algebraic/discrete. It contains a vast bibliography and a good history of the analysis of colored patterns.
[Ne] Needham, Tristan, Visual Complex Analysis, Clarendon Press, Oxford, 1997.
[Ma] Martin, George E., Transformation Geometry: An Introduction to Symmetry, Springer, New York, 1982.

This is a great place to read the proof of the fact that there are only 17 wallpaper groups.
[Sch] Schattschneider, Doris, "The plane symmetry groups. Their recognition and notation," Amer. Math. Monthly 85 (1978), pp. 439-459.

The table of fundamental cells of the 17 wallpaper groups is surely a contender for the most-photocopied page of any Monthly. The challenge of telling p31m from p3m1 [link] is well addressed.
[Sh] Shubnikov, A. V., N. V. Belov, and others, Colored Symmetry, Pergamon, New York, 1964.

This includes several of the original articles. It is a good place to read about Shubnikov's "rational" notation.
[Wa] Washburn, Dorothy, Symmetries of Culture: theory and practice of plane pattern analysis, University of Washington Press, Seattle, 1988.

Washburn is an anthropologist who has applied mathematical notation to the classification of patterns in art from around the world. There are excellent flow-charts for the classification of pattern types. An interesting history of this subject is a highlight. Also valuable is an extensive bibliography.
[Wo] Woods, H. J., "The Geometrical Basis of Pattern Design Part IV: Counterchange Symmetry in Plane Patterns," Journal of the Textile Institute, 27 (1936), pp. T305-T320.

Although appearing in a technical textile journal, this is the first place the two-color patterns were identified. It would be a good student exercise to relabel Woods' diagrams and examples with more modern notation. Though this journal is not widely available, we were very happy to find it at Brown University.

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Communications in Visual Mathematics, vol 1, no 1, July 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 8 Jul 1998 --- Last modified: Sep 30, 2003 4:46:44 PM
Comments to: CVM@maa.org