Tag Archives: Banach-Tarski Paradox

Paradoxical Playfulness

Q: What’s a good anagram for Banach-Tarski?

A: Banach-Tarski Banach-Tarski.


If you’re scratching your head about this one, don’t worry.  I was too at first.  I had to do some research about the Banach-Tarski Paradox before I understood the joke.  Now that I get it, I think it’s quite clever; sadly, it has very limited re-tell value.  (See what I did there?)

Let’s start with the easy part: an anagram is a rearrangement of the letters in a word or phrase, usually in such a way that they make readable new words.  For example, you can rearrange the letters in Nerdy Jokes to form Yo, Send Jerk!  I’m not sure what that means, but there you go.

You normally expect an anagram to have the same total number of letters as the original word or phrase, and that’s sort of where the humor of this jokes comes in.  Let’s talk about the Banach-Tarski Paradox, and maybe you’ll see why this joke is funny.

The Banach-Tarski Paradox comes from something called Set Theory: a mathematical theory dealing with the ancient Egyptian storm god, Set (I assume).  I’m not really sure how Egyptian mythology ties into the Banach-Tarski Paradox, but hey, I’m not a mathematician!

The Banach-Tarski Paradox starts off like this:  Imagine a three-dimensional solid ball.  Are you imagining it?  See how round it is?  Good.  Now imagine breaking the ball into a finite number of pieces.  Next, rearrange the pieces to form two solid daughter balls from the broken pieces of the original.  You can’t stretch or add pieces, you can only move and rotate the existing pieces.

Now…imagine that each daughter ball has the exact same volume as the parent.

If that doesn’t make you cry “Balderdash!” then perhaps you haven’t been paying attention.  I’ve just said that you can take the limited pieces of an object and rearrange them in such a way as to make two exactly identical copies.  That flies in the face of everything we think we understand about the relative permanence of volume.  You simply shouldn’t be able to do that.

But mathematically, you can.  And furthermore, you can do it endlessly, meaning that a single ball can be replicated endlessly until the universe is lousy with balls.

If you’re the enterprising sort, you’re probably already scheming about a way to get your hands on a gold bar and a hacksaw, perhaps thinking that you can replicate your wealth ad infinitum.  Well, here comes the part where I rain on your parade by telling you the caveats I didn’t mention before.

This only works on mathematical objects.  A mathematical object is different from a physical object in several important ways: first, a mathematical object is a collection of points (not particles), which, in the case of a ball, lie within well-defined parameters.  Second, since mathematical objects are not made of atoms, there’s no limit to how often or how finely they can be divided.  Third, and perhaps most importantly, the non-granular nature of mathematical objects means that when you do divide them into smaller pieces, the facets of the cut can be arbitrarily, even infinitely, small.

This potential for infinitely rough cuts makes the Banach-Tarski rearrangement possible.  A piece with an infinitely rough surface has an undefinable volume; consequently, the normal rules for adding volumes do not apply.  In fact, you only need to divide a mathematical sphere into five pieces to create two clones, which is somewhat suprising to me.  Of course, the whole idea is surprising to me, so I guess more surprises shouldn’t be surprising, and I think I’ve just talked myself into another paradox.

So let’s revisit the joke: a normal anagram of Banach-Tarski would yield a word or combination of words with a sum of twelve letters, since there were only 12 letters to start with (Example: Satan Hack Bit).  But a mathematical rearrangement of Banach-Tarski, in the manner of Banach-Tarski, would yield two identical copies of the original (assuming, of course, that Banach-Tarski were a mathematical collection of points instead of a real-world physical object.)  I have been assured that the mathematicians who first described the paradox, Stefan Banach and Alfred Tarski, were in fact physical objects, meaning they were not subject to the rearrangement that bears their name.  (Nerdy Jokes would like to strongly discourage its readers from attempting to perform Banach-Tarski rearrangements on living organisms.)