Symmetry-breaking in carpets may be categorized as transformations of color, shape, space, and pattern. Transformations of color include binary color change or color alternation, algorithmic color change, and random color change. Transformations of shape include arbitrary changes of shape (i.e. reduction in size or scale), the addition of other shapes, or a change in orientation. Transformations of space include the illusionistic treatment of space as by creating a perception of overlapping planes in two dimensions, or by the representation of illusionary interlace. These methods play with inherent ambiguities in pattern and tease our perception. Transformations of pattern include the abutment of border patterns with horizontal or vertical reflection indicating a change of symmetry while retaining form, or the arbitrary cut-off of a pattern by another pattern or border, and the juxtaposition of patterns.

Viewed as art, patterns with symmetry and symmetry-breaking are interesting for they delight as they confound. Symmetry in nature is always approximate. In the man-made world, patterns that rely on strict symmetry are boring. This is true not only for the viewer, but also for the maker (see WASHBURN and CROWE, 1988). Through the analysis of symmetry and symmetry-breaking in Oriental carpets, I feel that I have gotten closer to the minds of the makers - they were never bored! While symmetry may be a constraint in pattern-making, symmetry-breaking in art may fall on the side of choice.

The process of weaving a carpet, knot by knot, results in a fascinating relationship between numbers and patterns that is logical, predictable, and mathematically based. These relationships are inherent to the temporal processes of pattern formation. Both arithmetic and geometry are at once present, operating conjointly. They may be ignored on the part of the weaver, or played with purposefully to draw out inherent ambiguities in patterns. The grid of knots, side by side and above one another, is predicated upon the underlying interlacing of warp and weft. But the placement of color in repeated sequences thus sets up a series of relationships of corresponding points such that a plane pattern is established in which circles and centers are implied theoretically by the layout of the pattern (ALEXANDER, 1993). Rug-weaving is at once a unitary process, accomplished knot by knot, and a systemic process that results in a multiplicity of patterns effected by choice on the part of the weaver. While patterns in nature result from forces and constraints, patterns in rugs are the result of choices and constraints. Symmetry offers possibilities for the weavers, which are at once choices and constraints. While the possibilities for the composition of a design are limitless, once a weaver chooses to manipulate that design to create a pattern, the laws of symmetry limit those possibilities (see STEVENS, 1981). Patterns are restricted by the laws of symmetry - unless they are broken.

Although mathematicians treat symmetry as an ideal, in nature all symmetry is approximate. The study of patterns in 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. These carpets attest to a high degree of human creativity and ingenuity, but I think they express a genuine appreciation of a beauty informed by form, pattern, and structure. The study of patterns and pattern formation in Oriental carpets provides insights into the nature of beauty, which relies upon the beauty of nature in the realm of human choice.

References

1. ALEXANDER, C. (1993) A Foreshadowing of 21st Century Art. The Color and Geometry of Very Early Turkish Carpets, New York and Oxford.
2. BEATTIE, M. H. (1983) On the making of carpets, in Eastern Carpet in the Western World (eds. D. King and D. Sylvester), Arts Council of Great Britain, London, pp. 106-109.
3. BIER, C. (1992) Elements of plane symmetry in Oriental carpets, The Textile Museum Journal, 31, 53-70.
4. BIER, C. (1996) Approaches to understanding Oriental carpets, Arts of Asia, 26/1, 66-81.
5. BIER, C. (1997) Symmetry and Pattern: The Art of Oriental Carpets, <http://forum.swarthmore.edu/geometry/rugs/> The Math Forum at Swarthmore College and The Textile Museum.
6. MEINHARDT, H. (1995) The Algorithmic Beauty of Sea Shells, Springer-Verlag, New York, Berlin, and Heidelberg.
7. STEVENS, P. S. (1981) Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions, The MIT Press, Cambridge, MA and London.
8. WASHBURN, D. K. and CROWE, D. W. (1988) Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, University of Washington Press, Seattle and London.