There is an even stronger version of the paradox:

**Any two bounded subsets (of 3-dimensional Euclidean space****R**^{3}) with non-empty interior are equi-decomposable.

It is interesting to note that this theorem depends on three dimensions; while intuitively the two-dimensional case seems to be easier, it is in fact not true that all bounded subsets of the plane with non-empty interior are equi-decomposable. Still, there are some paradoxical decompositions in the plane: a circle can be cut into finitely many pieces and reassembled to form a square of equal area; see Tarski's circle-squaring problem.

Note that in the decomposition, the pieces won't be measurable, and so they will not have "reasonable" boundaries nor a "volume" in the ordinary sense. It is impossible to carry out such a disassembly physically because disassembly "with a knife" can create only measurable sets. This pure existence statement in mathematics points out that there are many more sets than just the measurable sets familiar to most people.

In 1924, Stefan Banach and Alfred Tarski described this paradox,
building on earlier work by Felix Hausdorff who managed to "chop up" the unit interval into countably many pieces which (by translation only) can be reassembled into the interval of length 2. He did this in order to show that there can be no non-trivial translation invariant measure on the real line which assigns a size to *all* sets of real numbers.

Logicians most often use the term "paradox" for a statement in logic which creates problems because it causes contradictions, such as the Liar paradox or Russell's paradox. The Banach-Tarski paradox is not a paradox in this sense but rather a proven theorem; it is a paradox only in the sense of being counter-intuitive. Because its proof prominently uses the axiom of choice, this counter-intuitive conclusion has been presented as an argument against adoption of that axiom.

A sketch of the proof follows. Essentially, the paradoxical decomposition of the ball is achieved in four steps:

- Find a paradoxical decomposition of the free group in two generators.
- Find a group of rotations in 3-d space isomorphic to the free group in two generators.
- Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
- Extend this decomposition of the sphere to a decomposition of the solid unit ball.

The group **G** can be "paradoxically decomposed" as follows: let `S`(`a`) be the set of all strings that start with `a` and define `S`(`a`^{-1}), `S`(`b`) and `S`(`b`^{-1}) similarly. Clearly,

**G**= {ε} ∪`S`(`a`) ∪`S`(`a`^{-1}) ∪`S`(`b`) ∪`S`(`b`^{-1})

**G**=`a``S`(`a`^{-1}) ∪`S`(`a`), and**G**=`b``S`(`b`^{-1}) ∪`S`(`b`)

In order to find a group of rotations of 3-d space that behaves just like (or "isomorphic to") the group **G**, we take two orthogonal axes and let `A` be a rotation of arccos(1/3) about the first and `B` be a rotation of arccos(1/3) about the second.
(This step cannot be performed in two dimensions.)
It is somewhat messy but not too difficult to show that these two rotations behave just like the elements `a` and `b` in our group **G**. We'll skip it. The new group of rotations generated by `A` and `B` will be called **H**. Of course, we now also have a paradoxical decomposition of **H**.

Step number 3: The unit sphere `S`^{2} is partitioned into "orbits" by the action of our group **H**: two points belong to the same orbit if and only if there's a rotation in **H** which moves the first point into the second.
We can use the axiom of choice to pick exactly one point from every orbit; collect these points into a set `M`. Now (almost) every point in `S`^{2} can be reached in exactly one way by applying the proper rotation from **H** to the proper element from `M`, and because of this, the paradoxical decomposition of **H** then yields a paradoxical decomposition of `S`^{2}.

(This sketch glosses over some details.
One has to be careful about the set of points on the sphere which happen to lie on an axis of rotation of some matrix in **H**. On the one hand, there are countably many such points so they "do not matter," and on the other hand it is possible to patch up even those points.)

Finally, connect every point on `S`^{2} with a ray to the origin; the paradoxical decomposition of `S`^{2} then yields a paradoxical decomposition of the solid unit ball (minus the origin, but that can be dealt with easily).

- Layman's Guide to the Banach-Tarski Paradox (from Kuro5hin)
- Francis E. Su: "The Banach-Tarski Paradox", http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
- S. Wagon: "The Banach-Tarski Paradox", Cambridge University Press, 1986