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- Tarski's_circle-squaring_problem abstract "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.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.It follows from a result of Wilson (2005) that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only.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. However, in the plane, a decomposition into finitely many pieces must preserve the sum of the Banach measures of the pieces, and therefore cannot change the total area of a set (Wagon 1993).".
- Tarski's_circle-squaring_problem wikiPageExternalLink books?id=_HveugDvaQMC&pg=PA169.
- Tarski's_circle-squaring_problem wikiPageExternalLink b44h1her.pdf.
- Tarski's_circle-squaring_problem wikiPageID "216811".
- Tarski's_circle-squaring_problem wikiPageRevisionID "600390057".
- Tarski's_circle-squaring_problem hasPhotoCollection Tarski's_circle-squaring_problem.
- Tarski's_circle-squaring_problem subject Category:Discrete_geometry.
- Tarski's_circle-squaring_problem subject Category:Euclidean_plane_geometry.
- Tarski's_circle-squaring_problem subject Category:Geometric_dissection.
- Tarski's_circle-squaring_problem subject Category:Mathematical_problems.
- Tarski's_circle-squaring_problem type Abstraction100002137.
- Tarski's_circle-squaring_problem type Attribute100024264.
- Tarski's_circle-squaring_problem type Condition113920835.
- Tarski's_circle-squaring_problem type Difficulty114408086.
- Tarski's_circle-squaring_problem type MathematicalProblems.
- Tarski's_circle-squaring_problem type Problem114410605.
- Tarski's_circle-squaring_problem type State100024720.
- Tarski's_circle-squaring_problem comment "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.".
- Tarski's_circle-squaring_problem label "Kwadratura koła Tarskiego".
- Tarski's_circle-squaring_problem label "Problema círculo-quadrado de Tarski".
- Tarski's_circle-squaring_problem label "Quadrature du cercle de Tarski".
- Tarski's_circle-squaring_problem label "Tarski's circle-squaring problem".
- Tarski's_circle-squaring_problem label "Квадратура круга Тарского".
- Tarski's_circle-squaring_problem label "塔斯基分割圓問題".
- Tarski's_circle-squaring_problem sameAs Quadrature_du_cercle_de_Tarski.
- Tarski's_circle-squaring_problem sameAs Kwadratura_koła_Tarskiego.
- Tarski's_circle-squaring_problem sameAs Problema_círculo-quadrado_de_Tarski.
- Tarski's_circle-squaring_problem sameAs m.01fr72.
- Tarski's_circle-squaring_problem sameAs Q926112.
- Tarski's_circle-squaring_problem sameAs Q926112.
- Tarski's_circle-squaring_problem sameAs Tarski's_circle-squaring_problem.
- Tarski's_circle-squaring_problem wasDerivedFrom Tarski's_circle-squaring_problem?oldid=600390057.
- Tarski's_circle-squaring_problem isPrimaryTopicOf Tarski's_circle-squaring_problem.