Seeing double

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Whenever I write a new blog post, I discover something new. Usually, it’s a fun maths fact, but this time I stumbled upon a website for aspiring millionaires. You’ll find there answers to all your burning questions about the most expensive things: celebrity homes for sale (it’s Billy Joel’s home in Sagaponack, New York, if you’re wondering), letters (Francis Crick’s explanation of the DNA molecule addressed to his 12-year-old son) or hotel suites in Paris (the royal suite in the Dorchester Collection’s Hôtel Plaza Athénée – I’ll try to persuade my university that it’s exactly where I should stay during my next visit to EDF, my French industrial partner).

All that information was almost useful (or at least provided me with a nice way to procrastinate), but today I’ll focus on a different object—a diamond football designed for the 2010 FIFA World Cup by South African jewellers Shimansky. 

What could be better than possessing such a jewel? Possessing two of them, for the same price! Don’t run to your local Lidl, however: it’s not South African week yet.

One of the coolest mathematical theorems is the so-called Banach–Tarski paradox. I’m very proud to say that Stefan Banach and Alfred Tarski were outstanding Polish mathematicians, the latter even graduated from my alma mater, the University of Warsaw.

So you can understand the first part of this theorem’s name—but why is it a ‘paradox’, not a ‘theorem’?

The paradox states that it’s possible to double the volume of a 3D object without adding extra points. No, it’s not a mistake, it’s not magic – it’s maths! This means that if you had the most expensive football on your desk, you could split it into a finite number of non-overlapping pieces, just to glue them back together, getting two identical copies of your ball. Please don’t try it before you finish reading though!

This theorem definitely contradicts our intuition, which is why we call it a paradox. Unfortunately, we can’t really apply it in the real, physical world. Not because the theorem doesn’t work, but because its assumptions don’t hold. The problem lies in physics, or more precisely in atoms.

The Banach–Tarski paradox takes advantage of the existence of non-measurable sets. These are collections of points that we can’t measure, not because our measuring devices aren’t good enough, but because we can prove that we just can’t define a sensible measure for such a set. It’s a bit like trying to measure the total weight of these cats—you’ll fail no matter how hard you try.

The key idea of Banach and Tarski was to divide the ball into a finite number of non-overlapping non-measurable pieces. Actually, five sets are enough (so you don’t even have to spend too much time cutting your precious ball), where one of them is the central point.

Can we do the same with the most expensive football? No, because all physical objects are measurable as they’re made up of finite collections of atoms—yes, you too. Don’t worry, I’m not going to leave you without a real-world example.

Take a balloon with some gas inside, release the gas into a container and fill two other balloons with the released gas – this will give you two balloons with identical volume, half of the volume of our initial balloon. Now halve the air pressure, so that both balloons double their size. You obtained two identical copies of the initial balloon without adding any extra atoms!

Hmm, but I told you that the Banach–Tarski paradox doesn’t work in the physical world… So I have to admit that I lied a bit in this experiment. Yes, the new balloons have the same volume, but the gas density inside is only half of the initial density, so they aren’t identical copies.

In my humble opinion, the Banach–Tarski paradox is one of the most mind-boggling maths facts (and here is one of the most ridiculous videos about it!). If you think the same, please leave a comment. But if it’s too much to handle, maybe treat yourself to the world’s most expensive coffee…