The Banach–Tarski paradox

26.07.10

I've come across several funky theorems in my time studying mathematics, but this one by far blows my mind the most. The theorem goes by the name of the Banach–Tarski paradox, and the reason it is called a paradox will soon become evident - believe me.

The theorem basically states that a solid ball in a 3-dimensional space can be split into a finite number of non-overlapping pieces and then be reassembled into two copies of the original ball. The reassembling can only be done by moving and rotating the pieces; no stretching or bending is allowed.



Now this immediately creates a paradox. In geometry, the act of splitting and rotating are volume preserving. This would make it impossible to end up with two units of volume when you only started with one unit. It is now we must ask the question; in what field of mathematics does this apply? And the answer is set theoretic geometry. Now, I have no thorough understanding of this subject but I will try to explain parts of the theorem in so called lay-mathematics' terms.

If two geometric figures can be transformed into each other, they are called congruent. In set theory we describe this by letting the geometric figures be a group (say X) and we let G be all the isometries of X. This can be extended to the idea of two sets being equidecomposable with respect to G, which means that both sets can be partitioned into the same finite number of respectively G-congruent pieces. Formally we can say if
$$A = \bigcup_{i=1} ^k A_i, \,\,\,\,\, B = \bigcup_{i=1} ^k B_i, \,\,\,\,\, A_i \cap A_j = B_i \cap B_j = \emptyset \,\,\,\,\, \forall i,j \,\,\,\, 1 \leq i < j \leq k,$$
and there are elements
$$g_1,...,g_k \in G$$
such that for each i between 1 and k,
$$g_i(A_i) = B_i,$$
then we say that A and B are G-equidecomposable using k pieces. Furthermore, if a set E has two disjoint subsets A and B such that A and E, as well as B and E, are G-equidecomposable then E is called paradoxical. Using this we can reformulate the theorem to the following:

A three-dimensional Euclidean ball is equidecomposable with two copies of itself.

This theorem is somewhat hard to believe, but there exist proofs. It only goes to show that we have to be extra careful when we define things such as "volume" in mathematics since our intuition often are contrary to what we will discover.

Exciting, in a way. :D