Mathematicians treat symmetry as an ideal. But it is in symmetry-breaking that beauty emerges from truth.

Symmetry-breaking requires an expectation of symmetry. Often, symmetry-breaking is confused with asymmetry. But, rather, asymmetry is simply the absence of symmetry. For symmetry-breaking to exist, a pattern must first be stablished. Symmetry then is expected, but somehow that expectation is not met.

There are many ways to effect symmetry-breaking in Oriental carpets. The study of Oriental carpets may lead one to suppose that in art, as in nature, it is in the approximation of symmetry, rather than in its precision, that beauty is to be found and appreciated.

What is an Oriental Carpet? No one ever set out to weave an Oriental carpet.

A woman, or a girl, or less often a young boy, simply set out to weave a carpet. The concept of Oriental carpet transcends original cultural origins; it is a Western construct. Typically, an Oriental carpet is oblong in format, with a central field surrounded by borders. Many different patterns are present.

All hand-made Oriental carpets are filled with symmetry-breaking.
 
 

All patterns, whether in nature or in art, exhibit a systematic organization. Symmetry offers several possibilities for the organization of a pattern, each of which results in sets of corresponding points. In rug weaving, the symmetrical repetition of a design to form a pattern is effected by counting and repeating sequences of knots.

Possibilities for the composition of a design are limitless, and rely upon choice. But possibilities for the repetition of a design are limited by the laws of pattern formation, subject to the constraints of symmetry.
 
 

There are many ways a weaver may choose to break the symmetry of a pattern. I have characterized twelve methods, which group into four categories of transformation:

These often appear in combination with one another. Typically, in Oriental carpets, there are numerous examples of symmetry-breaking in any given carpet.
 
 

Carol Bier is Research Associate at The Textile Museum in Washington, DC, where she had served as Curator for Eastern Hemisphere Collections for seventeen years from 1984 to 2001. She is author of The Persian Velvets at Rosenborg (Copenhagen, 1995), editor and contributing author of Woven from the Soul, Spun from the Heart: Textile Arts of Safavid and Qajar Iran (16th-19th Centuries) (Washington, 1987), and editor of The Textile Museum Journal. Her on-line exhibition, "Symmetry & Pattern: The Art of Oriental Carpets," at <http://forum.swarthmore.edu/geometry/rugs/>, has won numerous awards. It is a collaborative project of The Math Forum at Swarthmore College and The Textile Museum.