Tarski's circle-squaring problem is the challenge, posed by

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

of equal

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

. This was proven to be possible by

Miklós Laczkovich Miklós Laczkovich (born 21 February 1948) is a Hungarian mathematician mainly noted for his work on real analysis and geometric measure theory. His most famous result is the solution of Tarski's circle-squaring problem in 1989.Ruthen, R. (198 ...

in 1990; the decomposition makes heavy use of the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

and is therefore

non-constructive In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...

. Laczkovich estimated the number of pieces in his decomposition at roughly 10⁵⁰. 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. More recently, gave a completely constructive solution using about

10^

Borel pieces. In 2021 Máthé, Noel and Pikhurko improved the properties of the pieces. In particular,

Lester Dubins Lester Dubins (April 27, 1920 – February 11, 2010) was an American mathematician noted primarily for his research in probability theory. He was a faculty member at the University of California at Berkeley from 1962 through 2004, and in retirem ...

, 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 idealized pair of scissors (that is, having

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...

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 geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

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 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 The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...

; those decompositions can even change the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...

of a set. However, in the plane, a decomposition into finitely many pieces must preserve the sum of the

Banach measure In the mathematical discipline of measure theory, a Banach measure is a certain type of content used to formalize geometric area in problems vulnerable to the axiom of choice. Traditionally, intuitive notions of area are formalized as a class ...

s of the pieces, and therefore cannot change the total area of a set .

References

*. *. *. *. *. *. *{{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, a
p. 169
}. Discrete geometry Euclidean plane geometry Mathematical problems Geometric dissection

See also

References