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 ...
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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of equal
area
Area is the measure of a region's size on a 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 open surface or the boundary of a three-di ...
. It is possible, using pieces that are
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s, but not with pieces cut by
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 ...
s.
Solutions
Tarski's circle-squaring problem was
proven to be solvable 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. (1989 ...
in 1990. The decomposition makes heavy use of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
and is therefore
non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 10
50. The pieces used in his decomposition are
non-measurable subsets of the plane.
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 simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area.
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.
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
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
. More recently, Andrew Marks and Spencer Unger gave a completely constructive solution using about
Borel pieces.
Limitations
Lester Dubins,
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 ...
boundary).
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.
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 ...
; those decompositions can even change the
volume
Volume is a measure of regions in 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 mathematics, mathematical discipline of measure theory, a Banach measure is a certain way to assign a size (or area) to all subsets of the Euclidean plane, consistent with but extending the commonly used Lebesgue measure. While there are c ...
s of the pieces, and therefore cannot change the total area of a set.
See also
*
Squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
, a different problem: the task (which has been proven to be impossible) of constructing, for a given circle, a square of equal area with
straightedge and compass
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
alone.
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Discrete geometry
Euclidean plane geometry
Mathematical problems
Geometric dissection