Gracious Living

Banach-Tarski part 2
November 21, 2010, 23:30
Filed under: Algebra, Math | Tags: , , , , ,

So I sort of left you hanging last time.  We talked about equidecomposability, showed that F_2 was paradoxical under its own action on itself, and embedded F_2 into SO(3).  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.

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Banach-Tarski part 1
November 21, 2010, 06:33
Filed under: Algebra, Math | Tags: , , , , , ,

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 \mathbb{R}^3 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.”

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Vector Spaces
November 17, 2010, 17:30
Filed under: Algebra, Math | Tags: , , ,

Okay, it’s time to make a big leap forwards in terms of concreteness.  The Banach-Tarski paradox makes a strong statement about \mathbb{R}^3 that isn’t true about \mathbb{R}^2 or \mathbb{R}.  Now, we still don’t really know what \mathbb{R} is, but if we pretend we know what it is, we can say stuff about \mathbb{R}^3.  Certainly, \mathbb{R}^3 has the product topology of \mathbb{R}\times\mathbb{R}\times\mathbb{R} — 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 \mathbb{R}^3 is a vector space.

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