The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.

A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here, because in this case it is impossible to define the volumes of the considered subsets, as they are chosen with such a large porosity. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.

It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.

Banach and Tarski publication
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox:

Given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A1 ∪ ... ∪ Ak, B = B1 ∪ ... ∪ Bk, such that for each i between 1 and k, the sets Ai and Bi are congruent.

Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball A into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set B, which contains two copies of A.

The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed. The difference between the dimensions 1 and 2 on the one hand, and three and higher, on the other hand, is due to the richer structure of the group E(n) of the Euclidean motions in the higher dimensions, which is solvable for n = 1, 2 and contains a free group with two generators for n ≥ 3. John von Neumann studied the properties of the group of equivalences that make a paradoxical decomposition possible and introduced the notion of amenable groups. He also found a form of the paradox in the plane which uses area-preserving affine transformations in place of the usual congruences.

Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist. Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing Von Neumann conjecture.

Formal treatment
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembly. Its mathematical structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the notions of equidecomposable sets and paradoxical set. Suppose that G is a group acting on a set X. In the most important special case, X is an n-dimensional Euclidean space, and G consists of all isometries of X, i.e. the transformations of X into itself that preserve the distances, usually denoted E(n). Two geometric figures that can be transformed into each other are called congruent, and this terminology will be extended to the general G-action. Two subsets A and B of X are called G-equidecomposable, or equidecomposable with respect to G, if A and B can be partitioned into the same finite number of respectively G-congruent pieces. This defines an equivalence relation among all subsets of X. Formally, if there exist non-empty sets A1, ..., Ak, B1, ..., Bk such that
{\displaystyle A=\bigcup _{i=1}^{k}A_{i},\quad B=\bigcup _{i=1}^{k}B_{i},}
{\displaystyle \quad A_{i}\cap A_{j}=\emptyset ,\quad B_{i}\cap B_{j}=\emptyset \quad {\text{for all }}1\leq i<j\leq k,}
and there exist elements {\displaystyle g_{i}\in G} such that
{\displaystyle g_{i}(A_{i})=B_{i}{\text{ for all }}1\leq i\leq k,}
then it can be said that A and B are G-equidecomposable using k pieces. 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 terminology, the Banach–Tarski paradox can be reformulated as follows:
A three-dimensional Euclidean ball is equidecomposable with two copies of itself.
In fact, there is a sharp result in this case, due to R. M. Robinson: doubling the ball can be accomplished with five pieces, and fewer than five pieces will not suffice.

The strong version of the paradox claims:
Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable.
While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if A is equidecomposable with a subset of B and B is equidecomposable with a subset of A, then A and B are equidecomposable.

The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality. Thus, if one enlarges the group to allow arbitrary bijections of X then all sets with non-empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, in other words, by applying similarity transformations. Hence if the group G is large enough, G-equidecomposable sets may be found whose "size" varies. Moreover, since a countable set can be made into two copies of itself, one might expect that somehow, using countably many pieces could do the trick.