Transformations and SymmetrySymmetry Groups

Some shapes have more than one symmetry – let’s have a look at the square as a simple example.

You have already shown above that a square has axes of reflection.

It also has rotational symmetry by °, ° and °.

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 different “symmetries of the square”.

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

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

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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:

1. Adding two symmetries/integers always gives another symmetry/integer:
   +=
   12+7=19
   Continue
2. Adding symmetries/integers is associative:
   ++=++
   4+2+5=4+2+5
   Continue
3. Every symmetry/integer has an inverse, another symmetry/integer which, when added, gives the identity:
   +=
   4+–4=0
   Continue

In mathematics, any collection that has these properties is called a group. Some groups (like the 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…

The properties of the CCl₄ molecule (left) and the Adenovirus (right) are determined by their symmetries.