Tarski's circle-squaring problem
Encyclopedia
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 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. Such decompositions cannot be performed in the plane, due to the existence of a Banach measure
.
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
in 1925, to take a disc
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
of equal area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
. This was proven to be possible by Miklós Laczkovich
Miklós Laczkovich
Miklós Laczkovich 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.- Career :...
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
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
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
Paradoxical set
In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is a partitioning of the set into exactly two subsets, along with an appropriate group of functions that operate on some universe , such that each partition can be mapped back onto...
in three dimensions provided by the Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
; those decompositions can even change the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of a set. Such decompositions cannot be performed in the plane, due to the existence of a Banach measure
Banach measure
In mathematics, Banach measure in measure theory may mean a real-valued function on the algebra of all sets , by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round...
.
See also
- Squaring the circleSquaring the circleSquaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
, a different problem: the task (which has been proven to be impossible) of constructing, for a given circle, a square of equal area with straightedgeStraightedgeA straightedge is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines...
and compassCompass (drafting)A compass or pair of compasses is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as a tool to measure distances, in particular on maps...
alone.