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# Symmetries

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).    Next: Order of an element Up: MT2002 Algebra Previous: Permutations   Contents

Edmund F Robertson

11 September 2006