Suppose one has a rectangle and a rhombus neither of which are squares. Which is more symmetric?
I think that issues about symmetry naturally lead to ideas that show how powerful thinking about the theory of groups is.
In addition to the above question there is the issue of how many different kinds of patterns one can have on a strip (frieze patterns). Though visually there appears to be great complexity there is a natural sense in which there are only 7 types of patterns. One can generalize to patterns in the plane (17 wallpaper groups) and to patterns where colors are allowed.
Details and lovely pictures can be found in:
Grunbaum and Shephard, Tilings and Patterns Conway, Burgiel, and Goodman-Strauss, The Symmetries of Things