- Tarski's circle-squaring problem
Tarski's circle-squaring problem is the challenge, posed by
Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equalarea . This was proven to be possible byMiklós Laczkovich in 1990; the decomposition makes heavy use of theaxiom 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 arenon-measurable subset s.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. TheBolyai-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 thevolume of a set. Such decompositions cannot be performed in the plane, due to the existence of aBanach measure .See also
*
Squaring the circle , a different problem: the (proved to be impossible) task of constructing, for a given circle, a square of equal area withstraightedge and compass alone.References
* 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.
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