We have seen number-theoretical and combinatorial examples of groups. Now we look at some geometrical ones.
Below we list some facts about symmetries.
Now, if we are given a figure in the plane (i.e. a set of points, like a line, or a triangle or a square, etc.) we can consider those symmetries of the plane which map this figure onto itself. It is easy to see that these symmetries also form a group; this group is called the group of symmetries of . It is worth remarking here that if is a finite (bounded) figure, then it follows from 6) that every symmetry of is either a rotation or a reflection.
Groups of symmetries of infinite figures are also of interest. Here one often considers a repeating pattern which fills a plane, rather like a wallpaper patterns. It is possible to classify all these groups, and it turns out that there are precisely 17 of them.
One can also consider the symmetries of the 3-dimensional space, rather than the plane, and also symmetries of 3-dimensional figures. Here, the analogue of wallpapers are crystals, and the classification of all possible groups arising here (there of them) is a significant piece of information in the study of crystals (called crystallography).
Edmund F Robertson
11 September 2006