The Banach-Tarski paradox allows one to, through decomposition and reassembly, turn one three-dimensional ball into two without changing the individual pieces, apparently violating the additivity of volume in Euclidean three-dimensional space. In the previous post, a decomposition of the group F2, roughly the set of finite strings of the symbols "a" and "b", was shown to yield two copies of the same group when the pieces were "translated" in a certain sense.
In carrying over the properties of F2 into three-dimensional space, one treats the symbols a and b as rotations about axes in Euclidean three-dimensional space. Traditionally, the axes are considered to be the x- and z-axes of Cartesian coordinates. In fact, the necessity of a choice of axes is why a paradoxical decomposition can only occur in dimensions of three and above, and not in two. This is because, in F2, the strings ab and ba are distinct; following their respective paths on the Cayley graph yields two different points. In two dimensions, any two rotations about the origin are commutative, i.e. can be performed in either order with the same result. Since the noncommutativity of F2 cannot be carried over into two-dimensional space, the paradox is not possible there.
The rotations that correspond to a and b are taken to move through an angle of the inverse cosine of 1/3, or about 70.5°. This exact angle choice is unnecessary, but the angles chosen for a and b must be irrational multiples of a right angle. This is because no two linear combinations of them can be allowed to yield the same rotation; all linear combinations of the angles must be distinct. The purpose of this condition is to mimic a property of the group F2, namely that no two distinct simplified strings represent the same element, or, in other words, no two distinct paths (without retracing) lead to the same point on the Cayley graph.
Next, we briefly restrict our attention to the sphere (which, unlike the ball, does not include the interior area; the sphere is as the surface of the earth and the ball like the surface as well as the interior). The set F2, which we shall now consider a group of rotations, can act on any point p of the sphere. The set of points thus obtained, following any sequence of rotations (each corresponding to an element of F2) beginning at p, is called the orbit of p.
In this way, the entire surface of the sphere can be partitioned into an infinite set of these orbits, none of which overlap. Since a set of finite sequences is countable, F2 is as well. Since the number of points on the sphere is uncountably infinite, it follows that there are uncountably many of these orbits. Here the axiom of choice is invoked to select a single point from each orbit, and collect these into another set M.
The details involved in explicitly applying the decomposition of the sphere are too technical to consider here; the above steps were included to illustrate the use of the axiom of choice. The next step essentially brings about the paradoxical decomposition of the sphere by shifting M by the rotations a and b. Two copies of the sphere arise in a manner similar to that of F2.
Finally, the result is extended to the three-dimensional ball by performing the decomposition on a continuum of spheres of radii 0<r<R, where R is the original radius of the ball being considered. Each point p on the outer sphere can be paired to a point on any of the smaller spheres by projecting inward along the ray from p to the origin (see below). Clearly the union of all these spheres contains all the points of the ball, with the exception of the origin, O.
The final obstacle, therefore, is proving that the ball with its center removed can be decomposed and reassembled to form the entire ball. In fact, there are subtle difficulties in doing this that do not concern us here. Once this is done, the Banach-Tarski theorem is proven.
Following this technical formulation, it is enlightening to step back and consider the implications of the paradox. It is important to see that the decomposition above could not be applied to a physical object. The above procedure depends on the infinite divisibility of the ball, which an object composed of matter does not possess. Additionally, the pieces in the decomposition, though finite in number, are not "chunks" of the ball but infinite collections of points, and so are not physically continuous.
Though inapplicable to the physical world, the Banach-Tarski paradox helps to elucidate the fundamental differences between mathematical and physical space, and the wide-reaching consequences of assuming statements such as the axiom of choice.
In response to this and similar paradoxes that follow from the axiom of choice, there have been attempts to appropriately weaken the axiom of choice to an axiom which, though giving most of the same benefits, eliminates the paradoxes. One of these is called the axiom of countable choice, which limits the applicability of the axiom to countable sets. This avoids the Banach-Tarski paradox, but some set theoretical results are lost. In addition, the rather arbitrary restriction to countable sets seems inelegant, as it complicates the axiom, bringing in more concepts.
Also, some interesting work has been done since the Banach-Tarski paradox was published in 1924 that has extended the result. First, the final step of the proof above, in its original form, involved a total of 24 pieces. Through an alteration of the orbit scheme above, the number of pieces can be reduced to five.
Furthermore, the beginning and ending sets can be more general than simply a ball and two balls. Clearly, by a repetition of the above process, any (finite) number of balls can be produced by decomposition. It has even been shown that, if the original ball can be decomposed into an infinite number of pieces, one can obtain infinitely many copies of the ball, and even uncountably many. By allowing these decompositions, we can simply conjure up as many balls as we want from a single one!
In fact, the statement has been generalized even further to allow any bounded three-dimensional regions which are not "empty" to be broken up into a finite number of pieces and reassembled into any other of these regions.
The Banach-Tarski paradox is central in proving that there is no finitely additive measure in three-dimensional (and higher) spaces which agrees with the basic conception of volume. In one and two dimensions, there still is no countably additive measure that can be universally applied due to the existence of non-measurable sets (see again the Lebesgue measure series for an example of a measure in mathematics). The above are a few of the surprising geometric applications of the axiom of choice, showing how pervasive this assumption is, even beyond its native set theory.
Sources: http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf, http://www.bsu.edu/libraries/virtualpress/mathexchange/05-01/Coleman.pdf, Banach-Tarski Paradox at Wikipedia
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