Banach-Tarski!: Fun with some very weird math

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 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.

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