In mathematics, we often find unexpected connections between a mathematical discovery and a real world phenomenon. One such example is the connection between the study of symmetry – in mathematical terms, we call this “group theory” – and quantum mechanics, ie: the way subatomic particles like electrons behave. Let’s explore group theory before delving into the way it is connected to quantum mechanics.

“Group theory” – let’s call it the “mathematical study of symmetry” – originated when some very dedicated mathematicians, Lagrange and Gauss, started investigating permutations. A good illustration of a permutation is shuffling a card deck: a card deck has 52 cards, and if we want to rearrange the cards, we can shuffle them, by hand or using a machine.

Shuffling the cards makes it so the order of the cards is random – this is important for card games like poker because it ensures the game is fair.

Now, if you think about it, there are a few different types of shuffles:

-There is the “non-shuffle”, where we don’t actually shuffle the cards, but leave them in place.

-There is the “anti-shuffle” – if we take a deck of cards, shuffle them, then return them to the original order, this would be an “anti-shuffle.”

-And there is a double shuffle – if we shuffle cards once, then shuffle them again, this would be a double shuffle. It gives the same result as a single shuffle – since the cards are randomized – but the “double shuffle” comes from shuffling twice, not once.

Sounds simple enough?

This illustrates all the basics of what mathematicians study as permutations: in mathematical terms, the “non-shuffle” of a card deck is called the “identity permutation”: it leaves everything in its original place.

The “anti-shuffle” is an example of what mathematicians call the “inverse permutation”: it takes a shuffle and un-does it, so the cards are returned to their original place.

And the “double shuffle”, where we shuffle a pair of cards twice, is an example of a repeated permutation – a permutation that comes as a result of two or more permutations. Shuffling the cards twice results in a card order that could come from a single shuffling, so we can consider the “double shuffle” as a special kind of shuffle.

You can probably think of other examples of re-arranging items: say, in a seminar, if everybody gets up from their seat during a break, then returns to a new seat after the break, this would be another example of a permutation.

So we can start thinking of a permutation in more abstract terms – not just as a specific example of a re-ordering or re-arrangement, but as a concept on its own.

This is what mathematicians refer to as “group theory.” It turns out that group theory gives us the ideal language – the ideal framework – to discuss the behavior of electrons, as we shall see.