# Transformations and SymmetryWallpaper Groups

In the previous sections we have now seen two different kinds of symmetry, that correspond to two different transformations: rotations and reflections. But there is also a symmetry for the third kind of rigid transformation:

**Translational symmetry**

In addition to reflectional, rotational and translational symmetry, there even is a fourth kind: **glide reflections**

A pattern can have more than one type of symmetry. And just like for squares, we can find the

These groups don’t tell you much about how the pattern *looks* like (e.g. its colours and shapes), just how it is *repeated*. Multiple different patterns can have the same symmetry group – as long are arranged and repeated in the same way.

It turns out that, while there are infinitely many possible patterns, they all have one of just 17 different symmetry groups. These are called the **Wallpaper Groups**. Every wallpaper group is defined by a combination of translations, rotations, reflections and glide reflections. Can you see the

Unfortunately there is no simple reason why there are *17* of these groups. Proving it requires much more advanced mathematics…

Instead, you can try drawing your own repeated patterns for each of the 17 wallpaper groups:

#### Examples of other students’ drawings

The wallpaper groups were all about flat, two-dimensional patterns. We can do something similar for three-dimensional patterns: these are called crystallographic groups, and there are 219 of them!

In addition to translations, reflections, rotations, and glide reflections, these groups include symmetries like **glide planes** and **screw axes** (think about the motion when unscrewing a bottle).