# Transformations and SymmetrySymmetry Groups and Wallpapers

Some shapes have more than one symmetry – let’s have a look at the

You have already shown above that a square has

It also has rotational symmetry by

And finally, we can think about “doing nothing” as another special kind of symmetry – because the result is (obviously) the same as before. This is sometimes called the **identity**.

In total, we have found

Now we can actually start doing some arithmetic with these symmetries. For example, we can *add* two symmetries to get new ones:

Whenever you add two symmetries of a square, you get a new one. Here is a “symmetry calculator” where you can try it yourself:

Spend some time playing around with the symmetry calculator, and try to find any patterns. Can you complete these observations?

- Adding two rotations will always give
(or the identity). - Adding two reflections will always give
(or the identity). - Adding the same two symmetries in the opposite order
result. - Adding the identity
.

You might have realised already that adding **symmetries** is actually very similar to adding **integers**:

- Adding two
**symmetries**/**integers**always gives another**symmetry**/**integer**:+ = 12 + 7 = 19 - Adding
**symmetries**/**integers**isassociative :+ + = + + 4 + 2 + 5 = 4 + 2 + 5 - Every
**symmetry**/**integer**has an**inverse**, another**symmetry**/**integer**which, when added, gives the identity:+ = 4 + –4 = 0

In mathematics, any collection that has these properties is called a **group****symmetries** of a square) only have a finite number of elements. Others (like the **integers**) are infinite.

In this example, we started with the eight symmetries of the square. In fact, every geometric shape has its own **symmetry group**. They all have different elements, but they always satisfy the three rules above.

Groups appear everywhere in mathematics. The elements can be numbers or symmetries, but also polynomials, permutations, matrices, functions … *anything* that obeys the three rules. The key idea of *group theory* is that we are not interested in the individual elements, just in *how they interact with each other*.

For example, the symmetry groups of different molecules can help scientists predict and explain the properties of the corresponding materials.

Groups can also be used to analyse the winning strategy in board games, the behaviour of viruses in medicine, different harmonies in music, and many other concepts…

## Wallpaper Groups

In the previous sections we saw two different kinds of symmetry corresponding 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).