TWO FOR THE PRICE OF ONE!

R. Ramanujam, The Institute of Mathematical Sciences, Chennai

Is it possible to take a solid sphere (a "ball"), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere?
In other words, can you double the volume of the original sphere?
Obviously impossible isn't it? Doing this would violate physical laws. Why stop at two pieces, we can go on making more and more spheres out of one.
The Banach-Tarski Paradox (BTP), well-known among mathematicians, particularly among set theorists, states that it is indeed possible to do this!
Remember, what we find to be possible or impossible usually depends on what all we assume to be possible or impossible, and it is the job of the mathematician to relentlessly pursue the implications of our assumptions. The Banach-Tarski (BTP) Paradox shows the consequences of an assumption called the Axiom of Choice, and is part of a very old debate among mathematicians on whether this axiom should be admitted or rejected.
What does BTP really say?
By now, you are probably worried that there is perhaps some trick in the statement of the BTP. There isn't, it is very easy to state formally.  All you need to know is the language of basic set theory and some intuitive ideas about rotation etc.
A sphere is just a solid ball of a given radius. Think of a sphere as a set of points that lie within it. For simplicity, let's assume a radius of 1, so our sphere would be the set:
    S = {(x,y,z) | x^2 +y^2 +z^2 <= 1 }
This is to be read as, S is the set of all points for which I can choose values (x,y,z) such that the squared distance from the centre (0,0,0) or radius is (x2+y2+z2), which is less than or equal to 1.
Here, the points x,y,z etc are all real numbers. (We know real numbers, they come up all the time in school, with decimal representations that may run on for ever without ever repeating a pattern. Examples are 2/3, \sqrt{2}, \pi, ....)
The Banach-Tarski Paradox (BTP) states, basically, that it is possible to take S (as we've defined above), and cut it up into n disjoint (non-overlapping) pieces, where n is a finite number. Let us call the pieces A_1 , A_2 , ... A_n. Hence the union of these pieces must give back S itself. The paradox states that if we perform some (finite) sequence of operations on each of the A_i 's, we will end up with two copies of S.
It is a simple statement, isn't it? The operations are rotations and translations. Rotations are what you think they are, and translations are movements vertically or horizontally, shifting the sphere as a whole.
Mathematical spheres
Having got the precise statement, let us get back to our intuition about what is possible. It is very important to note a very important difference between S and any real, physical sphere: S is infinitely divisible. Mathematically speaking, S not only contains an infinite number of points, it is also infinitely dense in the sense that between any two points, you can find infinitely many other points, and so on. There are also no holes inside the sphere. This is not true of a physical sphere, as there are only a finite number of atoms in any given physical sphere; so a physical sphere is not infinitely divisible.
Interestingly and importantly,  the Banach-Tarski paradox does not apply to spheres that are not infinitely divisible, and hence to physical spheres we know about. 
In the mathematical sphere S, each of the "pieces" is so infinitely complex that they are not "measurable". In everyday language, we can understand this to mean that it is impossible to measure their volume.  Real spheres do have volume, and we cannot cut atoms into arbitrary shapes, let alone into infinitely complex shapes.
It is interesting to note that one corollary of this paradox is that you can take a sphere, cut it into n pieces, remove some of the pieces, and reassemble the remaining pieces back into the original sphere without missing anything. Obviously, this is again impossible with a physical sphere. 
Managing infinity
In Mathematics, infinite objects are very common, and they do not always behave in ways that we consider intuitive, so it is better to be careful about mixing up everyday language with mathematical terms. While the proof of the Banach Tarski theorem is beyond what we can do in this article, we can see something similar and analogous, but very simple.
Consider the question: can we duplicate the set of integers? That is, given a set of integers, can we split it into two subsets such that each set is as large as the given original set? Intuitively, this is impossible: how can a part of a whole be as large as the whole?  Again, among finite objects this is impossible, but quite easy to do with infinite objects. How do you see this?
Let N be the (infinitely large) set containing all the positive integers from 1 to infinity. We can split them up into two sets, E containing all the even integers, and O containing all the odd integers. Surely, E and O are much smaller than N? But indeed no, they are, in fact, the same size.
First, we take E, and rename each member of E so that a number x is renamed to x divided by two. What do we get? Call this set E'.  We now find that E' = N.  Similarly, we take each member y from O, and rename y to (y-1)/2. Call this set O'.  Lo and behold, we also find that O' = N.  We have just duplicated the set of integers using nothing more than just the original integers. We didn't even need to use infinitely-divisible strange objects to achieve this.
Back to spheres
Coming back to spheres, it is helpful to keep in mind that each of the pieces A_i are potentially infinitely complex so that they do not have any well-defined volume. Now, this whole paradox may seem remotely possible if we had, say, required 1,000,000 pieces to achieve our feat; there's intuitively more room for some magic if we had a million pieces to play with. However, the stunning point about this whole paradox is that we don't need more than five pieces to achieve this feat. And unlike our odd/even number example, we do not need to play tricks with renaming individual points of each piece; we can perform the miracle by merely using well-behaved operations like rotations and translations.  Furthermore, one of these pieces only needs to contain the single point at the centre of the sphere.
In other words, it is mathematically possible to cut S into a mere four pieces (if we disregard the one center point), and to reassemble two of these pieces into the original S, and reassemble the other two into a copy of S.
The catch, of course, is that each of these four pieces are so complex that they do not have any "measure" (i.e., their respective volumes are not well-defined), and that we do not know how to mathematically describe them other than the fact that they exist and exhibit the strange re-assembly property. In fact, it is quite possible that each of those pieces consists of isolated points spread out throughout the entire volume of the original sphere S.
We can try some kind of "physical" explanation. Think of the infinitely complex pieces A_1 , ... A_n as a kind of "atom clouds", which are gas-like, non-solid (immeasurable). A physical sphere has some crystalline structure: that is a specific geometric arrangement of atoms inside it. These "clouds" lack this "crystalline structure" (i.e., they are unmeasurable). But still, by suitable rearrangement of them using rotations and translations, we can form them into two identical spheres, with half the density of the original so that they do have the same "crystalline structure". In mathematical terms, they have the same topological structure. These two spheres are identical to S, except for having only half the density of S.  However, S is infinitely dense, and so are its pieces. But then the two spheres we created are still infinitely dense. Thus the two spheres are identical to S.
In this sense, there is no paradox here after all. We are merely seeing a logical consequence of mathematical sets like S being infinitely dense.  This is very similar to how we duplicated the set of positive integers, so they each become identical to the original set. It is logical that we can keep extracting more volume out of an infinitely dense, mathematical sphere S.
A postscript
One question you may have is this: how did you actually manage to find those infinitely complex "pieces" of the sphere S, which we called A_1, ..., A_n? We do not know of any method (algorithm) to actually construct these sets. By the Axiom of Choice, they exist, and we can indirectly infer some of their properties (such as their being infinitely complex in the way we want).
The big debate among mathematicians is on whether such sets which might be non-constructible should be mathematically allowed at all. But it is very useful for deriving many results, and most mathematicians tend to accept it. 
Can the BTP be derived without making this strong assumption?  Very likely not, though nobody as yet been able to show that it cannot be derived without this axiom.