The limits of mixology

I want to say a few words about the Brouwer fixed point theorem. This was one of the early achievements of algebraic topology — which sounds scary — but bear with me. We’ll get the mathematical definition out of the way first and then I’ll tell you why it’s cool.

The Brouwer fixed point theorem states that any continuous function sending a compact convex set onto itself contains at least one fixed point.

Cool, right? Eh… no. If you’re anything like me, you didn’t understand a single word of that. Mathematicians have a real talent for obscuring the interesting in the technical. Let’s translate this from precise gibberish to imprecise English by way of example.

Let’s say we have some Play-Doh and we fashion it into a disk. We’ll pretend that it’s two-dimensional so it’s perfectly flat. We now have a disk that’s made of Play-Doh particles and each particle is at a specific point in the disk (some are at the top, some are at the bottom, and so on).

Now, we can perform a number of operations on this disk. We can fold it, bend it, stretch it, rotate it, or crumple it. We can keep doing these things to it for as long as we like before eventually making it back into a disk.

The Brouwer fixed point theorem guarantees that after all is said and done, there is at least one particle in our Play-Doh disk that is in the exact same place in the disk as it was before we started messing around with it. And this holds as long as we don’t tear the Play-Doh apart and stick it back together again. (That would violate the “continuous function” part of the definition.)

But here’s the kicker — this theorem isn’t limited to two dimensions. Let’s take another example: Nothing brings people together like a good cup of tea and mathematicians are no different. When making tea, let’s say you add milk and start stirring it to mix it in.

The Brouwer fixed point theorem says that — even if you stir your heart out for hours on end — when you’re done, at least one particle in the cup will be in the exact same place as before as started stirring. It is impossible to stir it so that every particle is in a new position in the cup and there is nothing you can do about this. It’s unintuitive, but it’s a fact of the Universe.

Like the Play-Doh, this holds for operations like stirring, but not for operations such as shaking as, in the language of a mathematician, that wouldn’t be a continuous mapping.

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