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