Plane Patterns

Over the years, I have accumulated about 30 neckties. These strips of fabric come in a variety of colors and patterns. The older ones tend to be plainer. Most of them feature rather simple, symmetric patterns. My newer ties are splashier, with brightly colored splotches, wildly zigzagging lines, and unusual geometric shapes. Even those ties, however, were probably cut from a larger piece of cloth dyed or woven with a repeating pattern.

Lately, I have enjoyed creating my own (potential) necktie designs--experimenting with the delicate balance between perfect regularity and complete randomness that characterizes much contemporary textile design.

For those who like to control their artistic efforts with algorithmic precision, the Mathematica-based computer program Artlandia is one possible starting point (see Visiting Artlandia, July 20, 1998). I used it to create the following pattern, which has so-called p4g symmetry.

TILE

The p4g symmetry characterizes one of the 17 possible wallpaper patterns--repeating, or periodic, patterns that cover the whole plane. The International Crystallographic Union established the standard notation for these patterns. In this case, the cryptic notation specifies a square pattern with reflections and quarter turns. Here's what such a pattern, in its simplest form, looks like.

Given that my home computer isn't particularly powerful, it takes Artlandia a long time to calculate, point by point, what a given design element would look like embedded in different wallpaper patterns. That tends to discourage experimentation.

Now, however, I can create a design in Artlandia, convert it into a bit-mapped graphics file, import that file into an amazing, new program called KaleidoMania!, and quickly try out all sorts of symmetries. Here are two, rapidly-computed variations of a piece of my original Artlandia design.

Developed by Kevin D. Lee of Sandpiper Software in St. Paul, Minn., and destined for classroom use, KaleidoMania! isn't commercially available yet. However, Lee will be demonstrating his software at an upcoming workshop on Symmetries of Patterned Textiles. Organized by Dorothy K. Washburn of the Maryland Institute in Baltimore and Donald W. Crowe of the University of Wisconsin in Madison, the workshop will be held May 7-9 in Madison (see http://home.att.net/~dkwashburn/). It's designed to introduce both academic researchers and artisans to geometric symmetry.

One- and two-dimensional repeating patterns appear ubiquitously on surfaces, from quilts and colored fabrics to pottery and ceramic tiles, the workshop organizers note. Analyzing the symmetries that underlie such designs affords a way to systematically describe and compare pattern structures. It also provides new perspectives on the complex interactions between design, technology, and culture in different societies.

Here's an additional example of an Artlandia object (left) redeployed in patterns with different symmetries (middle, right). I'll have more to say about KaleidoMania! in a future article.

The February issue of Mathematics Magazine furnishes another striking example of the link between mathematical symmetry and artistic effort. Frank A. Farris of Santa Clara University in California and Nils Kristian Rossing of SINTEF Telecom and Informatics in Trondheim, Norway, describe mathematical recipes for weaving rope mats that feature particular symmetries (strip, or frieze, patterns).

As the philosopher-theologian Thomas Aquinas (1224-1274) remarked more than 700 years ago, "The senses delight in things duly proportional." Symmetry constitutes an important element of that aesthetic appreciation.

References:

Bakshee, I. 1998. Exploring Artlandia. Mathematica in Education and Research 7(No. 4):46.

Farris, Frank A., and N.K. Rossing. 1999. Woven rope friezes. Mathematics Magazine 72(February):32. (Additional information is available at http://www.maa.org/pubs/mm_supplements/farris/rope.html.)

GrŸnbaum, B., and G.C. Shephard.1987. Tilings and Patterns.New York: W.H. Freeman.

Hilton, P., D. Holton, and J. Pedersen. 1997. Mathematical Reflections: In a Room with Many Mirrors. New York: Springer-Verlag.

Pedersen, J. 1983. Geometry: The unity of theory and practice. Mathematical Intelligencer 5(No. 4):37.

Washburn, D.K., and D.W. Crowe. 1988. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle, Wash.: University of Washington Press.

Information on the "Symmetries of Patterned Textiles" workshop can found at http://home.att.net/~dkwashburn/.

Information about KaleidoMania! is available from Kevin Lee (kdlee@acm.org). Artlandia is featured at http://www.artlandia.com/.

You can learn more about wallpaper pattern symmetries at http://www.geom.umn.edu/education/math5337/Wallpaper/ and http://aleph0.clarku.edu/~djoyce/wallpaper/seventeen.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.