A Layman’s Explanation of the Banach-Tarski Paradox


The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.


Actually, regarding math topics, wiki often makes you more confused than you already were. But this one is not bad. (Not *too* bad.)

The Banach-Tarski Paradox as a topic was chosen by Patrick K, who attends SMU. The idea of the paradox is simply that you can double the volume of a 3-dimensional set of points without adding any new points. Why is it a paradox? Well, it defies intuition because in our everyday lives we normally never see one object magically turning into two equal copies of itself.

It’s because it’s not possible in our physical world. The mathematical version of the paradox uses the concept of an immeasurable set. Every object in real life is measurable, because it is the set of a finite number of atoms taking up a finite amount of space. Mathematically, even when finite becomes infinite, you still usually have measurable sets. You really have to try very hard in order to create an immeasurable set.

The Banach-Tarski paradox splits the sphere into a finite number of immeasurable sets of points. The key word is finite. In fact, it can be shown that it can be split into just FIVE pieces, one of them being the point at the center. So with the other four pieces, we can separate them into two groups of two, and create an entire sphere out of each group, each the same size as the original sphere.

Though this is impossible to do in real life (because we are bounded by atoms), it is possible to make a real life analogy. This analogy will require basic knowledge of the gas laws, namely, that pressure and volume are inversely related. Here we go.

Consider an easily stretchable balloon with some volume of gas inside it. Now release the gas into some container and divide the gas in container to fill two balloons. Each new balloon will have one-half the volume of the original. But we’re going to introduce a trick. We’ll reduce the pressure of the room by half. This causes the balloons to each expand to double its size, so that each is as big as the original. We have reconstructed the paradox!

But wait, you say! Even though each new balloon has the same volume as the original, it has only one-half the density. So they’re not the same balloon as the original.

That objection is correct for the physical world. But in mathematics, we CAN get two identical spheres out of one. Here’s the catch. The mathematical sphere has infinite density. When you cut an infinite density in half, the new density is still… infinity. This explains the paradox.

Also, to have this paradox, you need this thing called the Axiom of Choice. You can check out the wiki article on it if you want; however, prepare to encounter some real math.

7 thoughts on “A Layman’s Explanation of the Banach-Tarski Paradox

  1. I know you mentioned that the paradox does not apparently apply to our physical world, but I wonder whether or not these ideas can be connected in anyway to the idea that physical reality is continuous. I think I get the overall gist of the explanation mathematically, but what does this situation say about the status of continuous space and maybe time, in particular, the idea that space is made up of points and instants, assuming the same ideas apply? Does this mean that two objects can ultimately be composed of an equal infinity of points at a fundamental level, even though they have distinguished sizes on an approximate level?

    If so, would a continuous space (at least as defined here) thus be impossible at the risk of falling into such an odd result? Or, if this conception of the continuum is preserved, should we try to look at space and time in a different way (perhaps we can say that on an approximate scale, that our normal intuitions still apply, even though it does not apply at the fundamental level of points, if that makes any sense)?

    I don’t have much of a background on the maths behind the paradox, but I do have some concerns about its implications for the concept of continuity. I hope you can provide a simple answer here.


  2. That is a beautiful and thorough explanation. The type of work I had been searching for to have a good background understanding of amenable algebra. For my thesis on ‘Amenability of Banach Algebra’. Thank you a million times.


  3. This seems to me to be a fancy way of reasoning that a subset of infinity is still functionally infinite. It will be interesting to see if anything useful can come of it someday.


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