Possibilities for the composition of a design are limitless, and may rely upon choices. But possibilities for the repetition of that design, whether symmetrical or asymmetrical, are limited by the laws of pattern formation and are subject to the constraints of symmetry.

In all patterns there are four basic symmetry operations that may be performed upon a fundamental region, design or motif. Mathematicians call these rigid motions because they suggest movements without distortion of size or shape around a point, along or across a line, or to cover a plane.

Four Basic Patterns: Here, the letter F (and the blank space around it) is taken as our fundamental region to demonstrate the four basic symmetry operations or rigid motions:

In rug weaving, the repetition of a design to form a pattern is accomplished by counting and repeating sequences of knots. The basic symmetries in carpets are thus effected knot by knot. Border Patterns: In carpets, border patterns result when any or several of the basic symmetries are repeated in one direction. The constraints of symmetry are such that there are seven possible combinations. These are two of them:

Field Patterns: Result when symmetrical repetition takes place in two directions, thus forming a two-dimensional pattern that covers the plane. There are 17 systems which mathematicians classify as symmetry groups. Here are two of them:

Symmetry in nature is always approximate. In the manmade world, patterns that rely on strict symmetry are boring–just think of all the hotel carpets you have seen with symmetrical patterns. In my study of classical Oriental carpets, through the analysis of symmetry and symmetry-breaking, I feel that I have gotten closer to the minds of the makers–and they were never bored!

Mathematicians treat symmetry as an ideal state. But it is in symmetry-breaking that truth is transformed to beauty. 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 achievement, that beauty is to be found.--Carol Bier.