In this chapter we introduce and exemplify the division of plane patterns into 17 types by symmetry group. This begins with the broad division into net type. The chapter concludes with a scheme for identifying pattern types, plus examples and exercises. It then remains to show that all the types are distinct and that there are no more; this will be done in Chapter 6.

Preliminaries

Here we recapitulate on some important ideas and results, then introduce the signature system which will label each type of plane pattern according to its symmetry group. For the basics of a plane pattern F and its group of symmetries G, see Review 4.1. We have introduced the subgroup T of G, consisting of all translation symmetries of F (Definition 4.2), and the representation of those translations by a net N of points relative to a chosen basepoint O (Definition 4.3). The points of N are the vertices of a tiling of the plane by parallelogram cells (Construction 4.5 – see especially Figure 4.3).

The division of patterns into five classes according to net type (determined by T) is motivated by reflection issues in Section 4.3.1. In Section 4.3.3 we described the five types, indicating case by case which of the feasible rotational symmetries for a plane pattern (Section 4.3.2) are permitted by net invariance, Theorem 4.14.