Oftentimes we find that topics in theoretical mathematics, and real world phenomena, end up having unexpected connections. One example of this is the relationship between group theory – the study of symmetry – and quantum mechanics.

Quantum mechanics examines the behavior of subatomic particles. Electrons, and other subatomic particles, do not behave in quite the same way as objects in our normal day-to-day world.

For example, in the real world, if a person spins around one time, they will return to their original state. You can try this out – if you get up from your computer, and actually physically spin yourself around one time, you’ll be back in your original position, though perhaps slightly dizzy.

Likewise with cars – if a car drives around in a circle, and returns to the spot where it started, it will be the same car – maybe with a little less gas, but the same.

Electrons are not like this, though. If an electron spins around one time, it changes its electromagnetic structure. This is something like if the Earth’s north and south poles switched every time the Earth spins around: the Earth would then have to spin twice for its north and south poles to end up in the same position.

In the same way, an electron has to spin around 2 times – actually, 2.002 times, to be more precise – to arrive back in its original state.

Strange, like the Twilight Zone, huh?

And it turns out that group theory – the language of permutations – gives a natural way of describing how the electron behaves in this way. Group theory gives a way to model the electron’s “double spin” behavior in a way that cannot be done using more simple mathematics. Another way to think about this is: the way electrons behave is more like the way cards behave when they are shuffled, than the way billiard balls behave on the pool table.

The details are on the level of graduate mathematics, but the takeway point is that a branch of mathematics, developed nearly 300 years ago, turns out to be the perfect language for describing aspects of nature we have only recently discovered.

LaGrange and Gauss certainly had no way of foreseeing this – to them, the universe was seen as a gigantic billiard table, like a machine. It turns out that, at the subatomic level, the reality is far more complicated, but mathematics gives us a natural language to understand how it operates.