What is the Banach–Tarski paradox?

The Banach–Tarski paradox is a theorem in set-theoretic geometry. It says that a solid ball in three-dimensional space can be decomposed into a finite number of disjoint subsets, which can be moved around and rotated, without changing their shape, to form two identical copies of the original ball. It is called a paradox because it clashes with geometric intuition, but it is a veridical paradox: it is not false or self-contradictory. [1] [2]

Stefan Banach and Alfred Tarski published the construction in 1924. It built on earlier work by Giuseppe Vitali on the unit interval and by Felix Hausdorff on paradoxical decompositions of the sphere. MathWorld reports that the original decomposition used six pieces, and that Raphael Robinson reduced the number to five in 1947. Five is the minimum, although four pieces suffice if the single point at the centre is ignored. [1] [3]

A stronger form says that any two bounded sets in three or more dimensions with nonempty interiors can be cut into the same finite number of pairwise congruent pieces. This is the informal "pea and the Sun" version. The strong form fails in one and two dimensions. The difference comes from the group of Euclidean motions, which is solvable in dimensions one and two but contains a free group with two generators from three dimensions upward. [1]

Rotations and translations ought to preserve volume, and that principle is part of the formal definition of volume. The catch is that the pieces are not solid chunks but infinite scatterings of points, and their volumes cannot be defined. They are non-measurable sets with respect to Lebesgue measure. If they were all measurable, the two resulting balls would have to have the same total volume as the original, which is impossible. So the theorem does not contradict the usual rules for volume. [1] [3]

The proof depends on the axiom of choice, which permits the construction of non-measurable sets and requires an uncountable number of choices. For this reason the result is not a practical cutting procedure. The Harvey Mudd College summary adds that matter is not infinitely divisible, so a block of gold could not be treated this way, and that the pieces are too jagged to have a well-defined volume. [1] [4]

Work published in 2005 showed that the pieces can be chosen so that they move continuously into place without colliding. Separately, results by Leroy and by Simpson show that the paradox does not violate volume if one works with locales (point-free topology) instead of ordinary topological spaces. In that setting the pieces intersect in a way that carries hidden positive mass, and all subsets of Euclidean space can be measured. The nLab entry notes that the paradox may stem not only from the axiom of choice but also from the point-based concept of topological spaces. [1] [5]

Quanta Magazine describes the dissection as splitting the uncountably infinite set of points in a sphere into an uncountably infinite number of countably infinite sets, using a very specific procedure. [6]