We have seen number-theoretical and combinatorial examples of groups. Now we look at some geometrical ones.

Below we list some facts about symmetries.

- 1)
- Every symmetry is a bijection.
- 2)
- The composition of two symmetries is again a symmetry.
- 3)
- The inverse of a symmetry is again a symmetry.
- 4)
- The set of all symmetries is a group under composition of mappings.
- 5)
- A symmetry preserves angles.
- 6)
- Every symmetry is either a translation, or a rotation, or a reflection,
or a product of a translation and a reflection (called a
*glide-reflection*).

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