![Triangledissection](https://upload.wikimedia.org/wikipedia/commons/9/90/Triangledissection.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the Wallace–Bolyai–Gerwien theorem, named after
William Wallace
Sir William Wallace ( gd, Uilleam Uallas, ; Norman French: ; 23 August 1305) was a Scottish knight who became one of the main leaders during the First War of Scottish Independence.
Along with Andrew Moray, Wallace defeated an English army ...
,
Farkas Bolyai and
Paul Gerwien, is a theorem related to
dissections
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause of ...
of
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 to ...
s. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
and
rotations. The Wallace–Bolyai–Gerwien theorem states that this can be done
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
two polygons have the same
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 ope ...
.
Wallace had proven the same result already in 1807.
According to other sources, Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.
Formulation
There are several ways in which this theorem may be formulated. The most common version uses the concept of "equidecomposability" of polygons: two polygons are equidecomposable if they can be split into
finitely many triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s that only differ by some
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
(in fact only by a combination of a translation and a rotation). In this case the Wallace–Bolyai–Gerwien theorem states that two polygons are equidecomposable if and only if they have the same area.
Another formulation is in terms of
scissors congruence
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. 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 m ...
: two polygons are scissors-congruent if they can be decomposed into finitely many polygons that are pairwise
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
. Scissors-congruence is an
equivalence relation. In this case the Wallace–Bolyai–Gerwien theorem states that the
equivalence classes of this relation contain precisely those polygons that have the same area.
Proof sketch
The theorem can be understood in a few steps. Firstly, every polygon can be cut into triangles. There are a few methods for this. For
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
s one can cut off each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
in turn, while for
concave polygon
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.
Polyg ...
s this requires more care. A general approach that works for non-simple polygons as well would be to choose a
line not parallel to any of the sides of the polygon and draw a line parallel to this one through each of the vertices of the polygon. This will divide the polygon into triangles and
trapezoids
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eucli ...
, which in turn can be converted into triangles.
Secondly, each of these triangles can be transformed into a right triangle and subsequently into a
rectangle with one side of length 1. Alternatively, a triangle can be transformed into one such rectangle by first turning it into a
parallelogram and then turning this into such a rectangle. By doing this for each triangle, the polygon can be decomposed into a rectangle with unit width and height equal to its area.
Since this can be done for any two polygons, a "common subdivision" of the rectangle in between proves the theorem. That is, cutting the common rectangle (of size 1 by its area) according to both polygons will be an intermediate between both polygons.
Notes about the proof
First of all, this proof requires an intermediate polygon. In the formulation of the theorem using scissors-congruence, the use of this intermediate can be reformulated by using the fact that scissor-congruences are transitive. Since both the first polygon and the second polygon are scissors-congruent to the intermediate, they are scissors-congruent to one another.
The proof of this theorem is constructive and doesn't require 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 ...
, even though some other dissection problems (e.g.
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 Mikl ...
) do need it. In this case, 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.
Nonetheless, the number of pieces required to compose one polygon from another using this procedure generally far exceeds the minimum number of polygons needed.
Degree of decomposability
Consider two equidecomposable polygons ''P'' and ''Q''. The minimum number ''n'' of pieces required to compose one polygon ''Q'' from another polygon ''P'' is denoted by σ(''P'',''Q'').
Depending on the polygons, it is possible to estimate upper and lower bounds for σ(''P'',''Q''). For instance,
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 ...
proved that if ''P'' is convex and the
diameters
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
of ''P'' and ''Q'' are respectively given by d(''P'') and d(''Q''), then
:
If ''P
x'' is a rectangle of sides ''a''·''x'' and ''a''·(1/''x'') and ''Q'' is a rectangle of size ''a'', then ''P
x'' and ''Q'' are equidecomposable for every ''x'' > 0. An upper bound for σ(''P
x'',''Q'') is given by
:
Since σ(''P
x'',''Q'') = σ(''P''
(1/''x''),''Q''), we also have that
:
Generalisations
The analogous statement about
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
in three dimensions, known as
Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. 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 m ...
, is false, as proven by
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
in 1900. The problem has also been considered in some
non-Euclidean geometries
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
. In two-dimensional
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
and
spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
geometry, the theorem holds. However, the problem is still open for these geometries in three dimensions.
References
External links
Wallace–Bolyai–Gerwien TheoremScissors Congruence- An interactive demonstration of the Wallace–Bolyai–Gerwien theorem.
Video showing a sketch of the proofAn Example of the Bolyai–Gerwien Theoremby Sándor Kabai, Ferenc Holló Szabó, and
Lajos Szilassi
Lajos Szilassi (born 1942 in Szentes, Hungary) was a professor of mathematics at the University of Szeged who worked in projective and non-Euclidean geometry, applying his research to computer generated solutions of geometric problems. , the
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
A presentation about Hilbert's third problem at College of Staten Island CUNY- Abhijit Champanerkar.
*
Optimal dissection of a unit square in a rectangle
{{DEFAULTSORT:Wallace-Bolyai-Gerwien theorem
Euclidean plane geometry
Theorems in discrete geometry
Geometric dissection