Bolyai–Gerwien theorem
Encyclopedia
In geometry
, the Bolyai–Gerwien theorem, named after Farkas Bolyai
and Paul Gerwien, states that any two simple polygon
s of equal area
are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon.
"Rearrangement" means that one may apply a translation and a rotation
to every polygonal piece.
Unlike the generalized solution to Tarski's circle-squaring problem
, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
The theorem can be understood in two steps. First, every polygon can be cut into triangles – for convex polygons this is immediate, by cutting off each vertex in turn, while for concave polygons this requires more care – indeed right triangles, by dropping an altitude. This is sufficient to easily compute the area, as each right triangle is half a rectangle, or alternatively can be cut half-way up to be reassembled into a rectangle. The second and subtler step is that each right triangle (or equivalently rectangle) can be decomposed into a rectangle with a side of a given (unit) length. Once this is proven, it follows that every polygon can be decomposed into a rectangle with unit width and height equal to its area, which proves the theorem.
in three dimensions, known as Hilbert's third problem
, is false, as proven by Max Dehn
in 1900.
, father of János Bolyai
, first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace
had proven the same result already in 1807.
According to other sources, Farkas Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Bolyai–Gerwien theorem, named after Farkas Bolyai
Farkas Bolyai
Farkas Bolyai was a Hungarian mathematician, mainly known for his work in geometry, and of his son János Bolyai.-Biography:...
and Paul Gerwien, states that any two 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...
s 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...
are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon.
"Rearrangement" means that one may apply a translation and a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
to every polygonal piece.
Unlike the generalized solution to Tarski's circle-squaring problem
Tarski's circle-squaring problem
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...
, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
The theorem can be understood in two steps. First, every polygon can be cut into triangles – for convex polygons this is immediate, by cutting off each vertex in turn, while for concave polygons this requires more care – indeed right triangles, by dropping an altitude. This is sufficient to easily compute the area, as each right triangle is half a rectangle, or alternatively can be cut half-way up to be reassembled into a rectangle. The second and subtler step is that each right triangle (or equivalently rectangle) can be decomposed into a rectangle with a side of a given (unit) length. Once this is proven, it follows that every polygon can be decomposed into a rectangle with unit width and height equal to its area, which proves the theorem.
Higher dimensions
The analogous statement about polyhedraPolyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
in three dimensions, known as Hilbert's third problem
Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
, is false, as proven by Max Dehn
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
in 1900.
History
Farkas BolyaiFarkas Bolyai
Farkas Bolyai was a Hungarian mathematician, mainly known for his work in geometry, and of his son János Bolyai.-Biography:...
, father of János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
, first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace
William Wallace (mathematician)
William Wallace was a Scottish mathematician and astronomer who invented the eidograph.-Biography:Wallace was born at Dysart in Fife, where he received his school education...
had proven the same result already in 1807.
According to other sources, Farkas Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.
External links
- Wallace–Bolyai–Gerwien Theorem
- An Example of the Bolyai–Gerwien Theorem by Sándor Kabai, Ferenc Holló Szabó, and Lajos Szilassi, the Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
.