The Banach-Tarski paradox is one of the many places where higher-level math starts to sound like a stoned conversation in a Freshman college dorm room.
Imagine a ball. Now imagine cutting that ball up into a finite number of pieces. Six, maybe. Or five. The Banach-Tarski paradox proposes that you could take those pieces and, without stretching or expanding them in any way, use them to form two balls identical to the first. Basically, you've just created mass out of nothing. That is, to put it mildly, not supposed to be able to happen. Thus, the part about the paradox.
WTF, you may ask? It might help to know that Banach-Tarski isn't talking about real, physical balls. Rather, it deals with theoretical, mathematical spheres. Unlike a real ball, which only has so many atoms, a theoretical sphere can be divided up into an infinite number of pieces. Comparing different explanations of Banach-Tarski that I found online, the one that made the most sense to me stared off with this detail, and was written by "The Writer" a contributor to kuro5hin.org. He or she put together a layman's analogy that lowers the "WTF!!?" to a nice, calm, "wtf?"
So here's my proposed "intuitive" rationalization of it. I'll do it by way of an analogy with a physical sphere.
Let's forget for the moment the mathematical sphere S, which has infinite density. Let's consider a real, physical sphere B (for "ball"), also of radius 1. B is identical to S except that it consists of a finite (albeit large) number of atoms. The way these atoms are laid out in B is called the crystalline structure of B. (I.e., if you take B, or any physical object for that matter, and look at it under an electron microscope, you will see the atoms laid out in a fixed, regular pattern. That's called its "crystal lattice".) Usually, the crystalline structure is a simple geometric relationship between neighbouring atoms.
Notice that although the geometric relationship between atoms define its crystalline structure, the precise distance between atoms may vary. This leads to materials of different densities.
Now, we perform the equivalent of a Banach-Tarski decomposition on our physical sphere B: we "atomize" B into four spherical clouds of atoms, let's call them C1, C2, C3, and C4. (We'll ignore the central atom in B, just as in the mathematical version of this decomposition.) Let's assume that each of these clouds are sparse enough that they are gaseous, no longer solid by themselves (imitating the immeasurability of the mathematical pieces of S). Furthermore, let's say that the atoms in each of these clouds are laid out in a regular pattern, so that if we rotate C1 by some angle G, and put it together with C2 in the same spherical region, the atoms in both clouds line up into the same crystalline structure as B, except that now the distance between atoms is greater (to account for the missing atoms now in clouds C3 and C4). Similarly, assume we can do the same with C3 and C4: we just translate them away from the original spherical region of B so that they don't interfere with C1 and C2, and reassemble them into another sphere.
Now, we have successfully built two (physical!) spheres with the same radius as B, using only material from B itself. Each of the two spheres have the same crystalline structure as B. The only difference between these spheres and B is that they each have only half the density of B.
Anyway, this very long introduction to a mathematical concept is necessary so that you can enjoy the funny video at the top of this post. In it, an onslaught of infinitely multiplying oranges overrun the University of Copenhagen mathematics department while students cavort to a parody version of Duck Sauce's "Barbara Streisand." It is weird and wonderful (and possibly a little NSFW in parts) and I wish I knew more about the awesome people who made it.
Thank you, samurai!