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
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 sameresult.
- 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 = 19Continue
- Adding symmetries/integers is
associative: + + = + + 4 + 2 + 5 = 4 + 2 + 5Continue
- Every symmetry/integer has an inverse, another symmetry/integer which, when added, gives the identity:
+ = 4 + –4 = 0Continue
In mathematics, any collection that has these properties is called a
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…
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:
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).