In Chapter 1 we combined two isometries g, h to produce a third by taking their compositions gh (do g, then h) and hg. There is another way to combine two isometries, of great practical use in the context of plane patterns, and which we will introduce in Section 2.3. We begin by highlighting two geometrical ways to find the composition (or product) of isometries. The first was already used in the proof of Theorem 1.18.

Method 1

1. (A) Determine the sense of the composition from those of its parts (Remark 1.17).

2. (B) Determine the effect of the composition on two convenient points P, Q.

3. (C) Find an isometry with the right sense and effect on P, Q. This must be the one required by Theorem 1.10.

Notice that (C) is now made easier by our knowledge of the four isometry types (Theorem 1.18). This method can be beautifully simple and effective for otherwise tricky compositions, but the second approach, given by Theorem 2.1 and Corollary 2.2, is perhaps more powerful for getting general results and insights. With Theorems 1.15 and 1.16 it says that every isometry can be decomposed into reflections, and it tells us how to combine reflections.

Method 2 Decompose the given isometries into reflections, using the available freedom of choice, so that certain reflections in the composition cancel each other out. See Examples 2.3 to 2.7. We note for later:

Method 3 Use Cartesian coordinates (See Chapter 7).