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No. 3200:
The Banach-Tarski Paradox

by Andy Boyd

Today, two for the price of one. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

The year 1924 brought to light one of the strangest results in the history of math - a result so strange it worried mathematicians. It stemmed from the work of two young Polish mathematicians, Stefan Banach and Alfred Tarski, both of whom would go on to have very successful careers. What is now known as the Banach-Tarski paradox is easy to explain, but impossible to believe.

Stephan Banach
Stephan Banach   Photo Credit: Wikimedia

Alfred Tarski
Alfred Tarski   Photo Credit: Wikimedia

Imagine you have a styrofoam ball. I give you a knife and ask you to cut it into five pieces. You can then rearrange those pieces to make all sorts of shapes - how interesting those shapes are depends on how you cut the ball and how you put the pieces back together. I now challenge you with the following task. Can you cut the ball in such a way that when you rearrange the pieces you get two balls identical to the ball you started with? Not two smaller balls, but two balls of exactly the same size? And you can't stretch anything. Just cut and rearrange.


Banach-Tarski paradox
Banach-Tarski paradox   Photo Credit: Wikimedia

At this point you're probably thinking you haven't understood me. But you have. Cut, rearrange the pieces, and voila - two balls identical to the one you started with. Against all intuition it can be done, even though it seems as if we're creating something from nothing. To find out how, let's look a little closer.

First, the five pieces, or sets, that we need can't be constructed with a knife. They're far too complicated - so complicated that they can only be constructed mathematically. Second, the pieces rely in a profound way on infinity, which is known for giving rise to strange states of affairs. For example, imagine a hotel with an infinite number of rooms labeled 1, 2, 3 and so on. Clearly, if all the rooms are occupied there's no room for another guest. But not so fast. If we ask everyone to move to the room one number higher than the room they're in, then room number 1 becomes available. It seems as if we've created something from nothing.

And that's how the Banach-Tarski paradox comes about. We define the five pieces such that when we divide them into two groups, give the pieces in each group a little twist in just the right way, the pieces in each group line up and we get two identical balls.

It's an unsettling result even for mathematicians. Mathematics routinely deals with infinity. But if we're not careful infinity can lead us astray. That's why we set up precise rules to deal with it. What's so interesting about the Banach-Tarski paradox is that as strange as it is, it comes about from the rules most mathematicians accept and work with. And therein lies the dilemma. Do we accept the rules we have, which give us the Banach-Tarski paradox, or do we look for new rules? It may surprise you to know that we've largely chosen to keep the existing rules. That could change. But for now, it seems we can get two balls for the price of one.

I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.

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The ability to construct two identical balls relies on a foundational mathematical principle known as the Axiom of Choice, which has a long, interesting, and often controversial place in the history of mathematics.

The Banach-Tarski paradox is often expressed in terms of creating two identical balls, though the result that lies behind the paradox is far more general, allowing for the construction of any number of balls and shapes of differing sizes.

Unfortunately, a detailed understanding the Banach-Tarski paradox requires some degree of college level mathematics.

Banach-Tarski Paradox. From the Wikipedia website: Click here. Accessed January 8, 2019.