Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math

So I sort of left you hanging last time. We talked about equidecomposability, showed that was paradoxical under its own action on itself, and embedded into . From here, it just becomes a matter of putting all the steps together: first the sphere, then the ball minus its center, then the whole ball.

Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math, topology

Okay, here’s the moment you’ve been waiting for: the proof of the Banach-Tarski Paradox. Here’s what the paradox says:

**Theorem** (Banach-Tarski). *There are a finite number of disjoint subsets of whose union is the unit ball, and such that we can apply an isometry to each of them and wind up with disjoint sets whose union is a pair of unit balls.*

Or “we can cut a unit ball up into a finite number of pieces, rearrange them, and put them back together to make two balls.”

Okay, it’s time to make a big leap forwards in terms of concreteness. The Banach-Tarski paradox makes a strong statement about that *isn’t* true about or . Now, we still don’t really know what is, but if we pretend we know what it is, we can say stuff about . Certainly, has the product topology of — but it has much more than this. It has an origin, for instance, and a distance function, and a way to measure angles. The distance function, in turn, allows us to define spheres and isometries (i. e. distance-preserving maps), which are both part of the statement of Banach-Tarski. All of these are summarized by saying that is a **vector space**.