Editing Tarski's circle-squaring problem

{{Short description|Problem of cutting and reassembling a disk into a square}}
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'''Tarski's circle-squaring problem''' is the challenge, posed by [[Alfred Tarski]] in 1925,{{r|tarski}} to take a [[Disk (mathematics)|disc]] in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a [[square]] of equal [[area]]. It is possible, using pieces that are [[Borel set]]s, but not with pieces cut by [[Jordan curve]]s.

==Solutions==
Tarski's circle-squaring problem was [[mathematical proof|proven]] to be solvable by [[Miklós Laczkovich]] in 1990. The decomposition makes heavy use of the [[axiom of choice]] and is therefore [[Nonconstructive proof|non-constructive]]. Laczkovich estimated the number of pieces in his decomposition at roughly 10<sup>50</sup>. The pieces used in his decomposition are [[non-measurable subset]]s of the plane.{{r|laczkovich|laczkovich2}}

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.{{r|laczkovich|laczkovich2}}

It follows from a result of {{harvtxt|Wilson|2005}} that it is possible to choose the pieces in such a way that they can be moved continuously while remaining [[disjoint sets|disjoint]] to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only.{{r|wilson}}

A constructive solution was given by Łukasz Grabowski, András Máthé and Oleg Pikhurko in 2016 which worked everywhere except for a set of [[measure zero]].{{r|grabowski}} More recently, Andrew Marks and Spencer Unger gave a completely constructive solution using about <math>10^{200}</math> [[Borel set|Borel pieces]].{{r|marks}}

==Limitations==
[[Lester Dubins]], [[Morris Hirsch|Morris W. Hirsch]] & Jack Karush proved it is impossible to dissect a circle and make a square using pieces that could be cut with an [[Idealization (science philosophy)|idealized]] pair of scissors (that is, having [[Jordan curve]] boundary).{{r|dubins}}

==Related problems==
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.{{r|laczkovich|laczkovich2}}

These results should be compared with the much more [[paradoxical decomposition]]s 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 measure]]s of the pieces, and therefore cannot change the total area of a set.{{r|wagon}}

== See also ==
* [[Squaring the circle]], a different problem: the task (which has been proven to be impossible) of constructing, for a given circle, a square of equal area with [[Compass-and-straightedge construction|straightedge and compass]] alone.

== References ==
{{Reflist|refs=

<ref name=dubins>{{citation |last1=Dubins |first1=Lester |last2=Hirsch |first2=Morris W. |last3=Karush |first3=Jack |date=December 1963 |title=Scissor congruence |journal=[[Israel Journal of Mathematics]] |language=en |volume=1 |issue=4 |pages=239–247 |doi=10.1007/BF02759727 | doi-access= |issn=1565-8511}}</ref>

<ref name=grabowski>{{citation
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| last3=Pikhurko | first3=Oleg
| date=27 April 2022
| title=Measurable equidecompositions for group actions with an expansion property
| arxiv=1601.02958
| journal=[[Journal of the European Mathematical Society]]
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| doi=10.4171/JEMS/1189 | doi-access=free}}</ref>

*{{citation
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 | volume = 44
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<ref name=laczkovich>{{citation
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<ref name=laczkovich2>{{citation
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<ref name=marks>{{citation
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<ref name=tarski>{{citation
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<ref name=wilson>{{citation
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 | url = https://authors.library.caltech.edu/11927/1/WILjsl05.pdf
 }}</ref>

<ref name=wagon>{{citation|title=The Banach–Tarski Paradox|title-link= The Banach–Tarski Paradox (book)|volume=24|series=Encyclopedia of Mathematics and its Applications|first=Stan|last=Wagon|authorlink=Stan Wagon|publisher=Cambridge University Press|year=1993|isbn=9780521457040|at=[https://books.google.com/books?id=_HveugDvaQMC&pg=PA169 p. 169]}}</ref>

}}

[[Category:Discrete geometry]]
[[Category:Euclidean plane geometry]]
[[Category:Mathematical problems]]
[[Category:Geometric dissection]]