In Chapter 1 we talked about symmetries of the line. Similarity symmetry changes lengths between points of the geometric figure and thus is not an isometry. A symmetry of a geometric figure is an isometry which takes the figure onto itself. For example, reflection though any median is a symmetry of an equilateral triangle. See Figure 18.2 Figure 18.2.
Every isometry is a symmetry of some geometric figure.
If we take a geometric figure, often called the motif , and apply to it the isometry and its inverse making additional copies of the motif over and over again, then we obtain another geometric figure for which the initial isometry is a symmetry. We generated two different patterns. A pattern is a figure together with all its symmetries . We call the collection of all symmetries of a geometric figure its symmetry group . Figure 18.5 .