EXAMPLE 7 Symmetry Groups of Strip Patterns

Each of the strip patterns of Figure 19.11 on page 795 has a different group of symmetries. What do they have in common, and how do they differ?

The pattern of Figure 19.11a is preserved only by translations. If we let denote the smallest translation to the right that preserves the pattern, then the pattern is also preserved by (which we write as ), by , and so forth. Although the pattern looks the same after each of these translations by different distances, we can tell these translations apart if we number each copy of the motif and observe which other motif it is carried into under the symmetry.

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For instance, takes each motif into another one that is located two motifs to the right. The symmetry has an inverse among the symmetries of the pattern: the smallest translation to the left that preserves the pattern; and (which we write as ), , and so forth also are symmetries. The entire collection of symmetries of the pattern is

From this listing, you see that it is natural to think of the identity as being . All of the strip patterns are preserved by translations, so the symmetry group of each includes the subgroup of all translations in this list. We say that the group is generated by and we write the group as , where between the angle brackets we list symmetries that, in combination, produce all the group elements. The symmetry group of Figure 19.11e includes, in addition, a glide reflection and all combinations of the glide reflection with the translations. Doing two glide reflections is equivalent to doing a translation, which we express as . The glide is only “half as far” as the shortest translation that preserves the pattern. Check that . The symmetry group of the pattern is

The pattern of Figure 19.11c is preserved by vertical reflections at regular intervals. If we let denote reflections at a fixed particular location, the other reflections can be obtained as combinations of and . To get a handle on what each of the symmetries does, it helps to make a “simplified” copy of the strip (we use Ws), number fixed positions on the page, and identify individual copies of the Ws with letters ("invisible,” because the letters are not part of the pattern) as in the following:

The symmetries move the Ws among the numbered positions. Let be the reflection across the vertical line through the middle of position 3, and let be the translation that moves each W one square to the right. To familiarize yourself with the symmetries, write out the result of each of , , and (recall, this is followed by ; for simplicity, we can omit the operation sign between symmetries). In each case, where does end up? The symmetry group of the pattern, the list of all of the symmetries, is

This group is notable because not all its elements satisfy the commutative property that , which you are accustomed to for numerical operations (). In fact, we do not have , but instead . We can express this group compactly as

where we list the symmetries that generate the group and indicate any relations that hold between them.