Translations and nets

Review 4.1 We recapitulate on some basic ideas. An isometry of the plane is a transformation of the plane which preserves distances, and is consequently a translation, rotation, reflection or glide (by Theorem 1.18). We may refer to any subset F of the plane as a pattern, but in doing so we normally imply that F has symmetry. That is, there is an isometry g which maps F onto itself. In this case g is called a symmetry or symmetry operation of F. Again, a motif M in (of) F is in principle any subset of F, but we generally have in mind a subset that is striking, attractive, and/or significant for our understanding of the structure of F.

Since the symmetry g has the two properties of preserving distance and sending every point of F to another point of F, it sends M to another motif M′ of F, which we may describe as being of the same size and shape as M, or congruent to M. By now we have many examples of this situation. An early case is that of the bird motifs of Figure 1.2, mapped onto other birds by translations and reflections. We observed that the composition of two symmetries of F, the result of applying one symmetry, then the other, qualifies also as a symmetry, and so the collection of all symmetries of F forms a group G (see Section 2.5). We call G the symmetry group of F.