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:
In addition to reflectional, rotational and translational symmetry, there even is a fourth kind:
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).