In Chapters 1 and 2 we have classified plane isometries, discovered some important principles of how they combine, and made a first application to patterns whose symmetry group is either the dihedral group D2n or its rotation subgroup Cn. Before investigating plane patterns it is a logical and useful step to classify the 1-dimensional, or braid, patterns, be aware of their symmetries, and get a little practice in both recognizing and creating them.

Definition 3.1 We say ν is a translation vector of pattern F if Tν is a translation symmetry. Then a braid (band, frieze) pattern is a pattern in the plane, all of whose translation vectors are parallel. In particular, a and −a are parallel. We will usually call this parallel direction horizontal, and the perpendicular direction vertical. Other names used are longitudinal and transverse, respectively. A symmetry group of a braid is sometimes called a line group.

As noted in Section 1.1, we are investigating patterns which are discrete: they do not have translation or other symmetries which move the pattern by arbitrarily small amounts. Thus, amongst the collection of all translation symmetries of the pattern there is a translation Ta of least but not zero magnitude. Of course it is not unique, for example T−a has the same magnitude |a| as Ta. We rephrase an observation from the preliminary discussion of braids preceding Figure 2.7. It may be derived more formally from Theorem 3.3.