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Tarski's circle-squaring problem
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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 Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich's decomposition uses about 1050 different pieces.
In particular, it's impossible to dissect a circle and make a square using pieces that could be cut with scissors (i.e. having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets.
Laczkovich actually proved 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 both 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. Such decompositions cannot be performed in the plane, due to the existence of a Banach measure.
- Miklos Laczkovich: "Equidecomposability and discrepancy: a solution to Tarski's circle squaring problem", Journal fur die Reine und 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.
- Squaring the circle, a different problem: the (provably impossible) task of constructing, for a given circle, a square of equal area with straightedge and compass alone.