Mathematical Chocolate for Everyone?

Have you ever dreamt about eating a chocolate bar that just never ends? Of course you have, we’re only human after all. Well, mathematicians have actually shown that this very concept is actually possible using maths. Don’t get too excited just yet though, as you may have guessed, it’s not something that even the most dedicated chocolatiers will ever be able to do. Saying this though, the nature of sub-atomic particles is a lot less predictable than the what we’re used to. So all hope is not completely lost just yet, because in certain processes that take place, particles do actually create something from nothing. This isn’t to say we’re on the brink of harnessing this into macroscopic reality, in fact, beyond these sub-atomic levels as far as we know it is actually impossible, but it’s something nice to think about at least.

It’s called the Banach-Tarski paradox, and personally as far as paradox’s go, its earnt its place in my favourites. The theorem is set in a theoretical geometry and it says that mathematically, it is possible to divide up any object, then rearrange the separated pieces in such a way that they form two identical copies of that initial object. The theorem was first stated in 1924 by two mathematicians named Stefan Banach and Alfred Tarski, (1).

The theorem is made possible using a shape in Euclidean space that is composed of an infinite number of points. These points can be described by a series of rotations from a certain starting point. But of course to describe an infinite number of points, there would also have to be an infinite number of starting points. So from this, we have a two infinite sets of points that make up the whole object; a rotation set and a starting point set. Now, if all the points that are part of the rotation set are labelled with the first rotation that it took to get to that point, these can be split into four more sets. Rotations starting with; up, down, left and right. From here, we can see that there are now 5 separate infinite sets that make up this object. Remember, each of these divided sets is still an infinite set, because infinity divided by a finite number is still infinity.

Now we’re ready to show where the trickery comes into it. If we take the ‘right’ rotation set and for each point, add a rotation left as the first rotation, the two left and rights would actually cancel out. Eliminating these first two rotations then would actually leave us with an infinite set of rotations starting with left, up, down and even all the starting points. Just take a moment here to think about this, by rotating one of the original sets from before, it has actually created a new set which consists of four of our original sets.  Repeating this whole step with respect to the up rotation set, will then produce our left rotation set again as part of its new set. By then adding all these sets together, we are actually left with the original object and after performing another rotation using one of the original starting points set, all the components needed to construct our second copy!

If you think about this for a while though, it is obvious to see that it only really works because of the definition of infinity. And if you then think even longer, although it is theoretically possible, it does seem to imply that it would take an infinitely complex object and an infinite time to be able to actually recreate this paradox in reality, which is slightly depressing and inconvenient for chocolate lovers everywhere.

By Jamie McGowan, Physics 2nd Year

  1. Banach, Stefan; Tarski, Alfred (1924). Review at JFM. “Sur la décomposition des ensembles de points en parties respectivement congruentes” (PDF). Fundamenta Mathematicae 6: 244–277.
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