Gracious Living

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.”

The original statement was written in terms of equidecomposability.  Suppose a group G acts on a set X.  Then A,B\subset X are Gequidecomposable if there is a partition of A into sets A_i, of B into sets B_i, and a list g_i of elements of G, where in all cases, i ranges from 1 to the same finite number k, such that g_i(A_i)=B_i for each i. We call a set paradoxical under the G-action if it can be partitioned into two proper subsets such that it’s equidecomposable with both. We’ll write \sim or \sim_G for equidecomposability, which is clearly an equivalence relation.

So basically what we’re trying to show is that the unit ball is E(3)-equidecomposable with two unit balls (where E(3)=\mathbb{R}^3\rtimes SO(3) is the group of isometries of space). This makes the pair of unit balls E(3)-paradoxical.

Let’s start with a simple example. I claim that the unit circle in the plane is E(2)-equidecomposable with itself minus a point. To prove this, let r\in E(2) be an “irrational rotation” — 1 radian, or \sqrt{2} degrees, or some other rotation that isn’t expressible as a rational number of full rotations. In particular, as an element of E(2), r can’t have finite order. Let p be a point in the circle. Then p,rp,r^2p,r^3p,\dotsc must all be different points (since r^k is a rotation, if it fixes one point, it must fix them all, which contradicts r being of infinite order). If we apply r to \{p,rp,r^2p,\dotsc\}, we get \{rp,r^2p,r^3p,\dotsc\}, which is the original set minus a point. Then just apply the identity to the rest of the circle, and the union of the two maps is a map from the circle to the circle minus a point, fitting the definition of equidecomposability above.

We’re actually going to use this later — in particular, it shows that any set in the plane or space that contains a circle is equidecomposable to itself minus a point.

Okay, now let’s get a little more abstract. Remember when we talked about free groups? It turns out that the free group on two generators, F_2, is paradoxical under its action on itself. Here’s why. Let W(a) be the set of reduced words starting with the generator a, and similarly for W(b),W(a^{-1}),W(b^{-1}).  Then these four sets and the singleton set \{e\} partition F_2.  But aW(a^{-1}) is the set of all words that <em>don’t</em> start with a, so aW(a^{-1})\cup W(a)=F_2.  Similarly, bW(b^{-1})\cup W(b)=F_2.  So this is a paradoxical decomposition of F_2.

Or, okay, it’s not, but it almost is, because I conveniently forgot about e above.  But this doesn’t break everything — just add it to one of the other sets, say W(a^{-1}).  And since we want the sets to stay disjoint, let’s also add a,a^2,a^3,\dotsc to W(a^{-1}).  Then aW(a^{-1}) is disjoint from W(a), bW(b^{-1}) is disjoint from W(b), and each pair partitions F_2.

But, see, we can apply this directly to the unit ball — because there’s an embedding of F_2 into SO(3).  There are different ways of doing this, but the basic idea is to take two irrational rotations about different axes and show they satisfy no relations.  I like Terry Tao’s smooth proof for this part of the argument.  Let a(x,y,z)=(\frac{3}{5}x+\frac{4}{5}y,\frac{-4}{5}x+\frac{3}{5}y,z),b(x,y,z)=(x,\frac{3}{5}y+\frac{4}{5}z,\frac{-4}{5}y+\frac{3}{5}z).  By writing down these guys in matrix form, you can check that they’re special orthogonal, and also that their inverses are given by just inverting the signs on the \frac{4}{5}s.  The trick now is to look at everything modulo 5.  See, if a and b have a nontrivial relation, then it translates to a word in 5a and 5b whose coefficients of x,y,z are all divisible by 5, so if we can show that one of these doesn’t exist, we’re done.  So we work over the finite field \mathbb{F}_5, which is \mathbb{Z}_5 additively and the multiplication is induced from that.  Mod 5, 5a(x,y,z)=(3x+4y,-4x+3y,0),5b(x,y,z)=(0,3y+4z,-4y+3z).  It’s pretty clear that the kernels of these maps are 2-dimensional vector spaces mod 5: the kernel of 5a, for instance, is $latez \langle (0,0,1),(3,4,0)\rangle$.  By a theorem of linear algebra, their images are thus 1-dimensional.  As it turns out, the images are perpendicular, so there’s no nontrivial relation in \mathbb{F}_5.

Later today, I’ll bring this all together.  Sorry this was late!

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[…] part 2 21 11 2010 So I sort of left you hanging last time.  We talked about equidecomposability, showed that was paradoxical under its own action on […]

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[…] of the big lessons learned from the Banach-Tarski paradox is that even in something as simple as a unit ball, we can find sets of impossible or […]

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