# Transformations and SymmetrySymmetry Groups

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
a rotationa reflection (or the identity). - Adding two reflections will always give
a rotationa reflection (or the identity). - Adding the same two symmetries in the opposite order
sometimes gives a differentalways gives a differentalways gives the same result. - Adding the identity
doesn’t do anythingreturns a reflectionreturns the opposite .

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…