Symmetries

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

Figure: Translation by the vector $\vec{a}$.

Figure 2: Rotation about the point $P$ by the angle $\alpha$.

Figure 3: Reflection in the line $l$.

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 $F$ 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 $F$. It is worth remarking here that if $F$ is a finite (bounded) figure, then it follows from 6) that every symmetry of $F$ is either a rotation or a reflection.

Figure 4: Symmetries of an equilateral triangle.

Figure 5: Symmetries of an isosceles triangle.

Figure 6: Symmetries of a rectangle.

Figure 7: Symmetries of a regular hexagon.

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 $230$ of them) is a significant piece of information in the study of crystals (called crystallography).