Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a circle (including its interior) in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklos Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore not explicit. Laczkovich's decomposition uses about 10⁵⁰ different pieces.

Laczkovich actually proved more: the reassembly can be done using translations only; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area. The Bolyai-Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if translations and rotations are allowed for the reassembly.

These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach-Tarski paradox; those decompositions can even change the volume of a set and cannot be performed in the plane.

References

• Miklos Laczkovich: "Equidecomposability and discrepancy: a solution to Tarski's circle squaring problem", Crelle's Journal of Reine and Angewandte Mathematik 404 (1990) pp. 77-117
• Miklos Laczkovich: "Paradoxical decompositions: a survey of recent results." First European Congress of Mathematics, Vol. II (Paris, 1992), pp. 159-184, Progr. Math., 120, Birkhäuser, Basel, 1994.

See also

• Squaring the circle is a different problem: the (provably impossible) task of constructing, for a given circle, a square of equal area with ruler and compass alone. Tarski's problem makes use of the (unprovable) axiom of choice to split the circle into large numbers of non-measurable subsets. As such, it cannot be implemented with physical tools, which can only draw measurable sets.