Tarski's Circle-Squaring Problem
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High Quality Content by WIKIPEDIA articles! 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 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 is impossible to dissect a circle and make a square using pieces that could be cut with scissors (that is, having…mehr

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High Quality Content by WIKIPEDIA articles! 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 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 is impossible to dissect a circle and make a square using pieces that could be cut with scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets.