Hilbert's third problem
Encyclopedia
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 second? Based on earlier writings by Gauss
, Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn
, who proved that the answer in general is "no" by producing a counterexample.
The answer for the analogous question about polygon
s in 2 dimensions is "yes" and had been known for a long time; this is the Bolyai–Gerwien theorem
.
,
had been known to Euclid
, but all proofs of it involve some form of limiting process
or calculus
, notably the method of exhaustion
or, in more modern form, Cavalieri's principle
. Similar formulas in plane geometry can be proven with more elementary means. Gauss regretted this defect in two of his letters. This was the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods? Because if not, then an elementary proof of Euclid's result is also impossible.
is used to prove an impossibility result in geometry
. Other examples are doubling the cube
and trisecting the angle.
We call two polyhedra scissors-congruent if the first can be cut into finitely many polyhedral pieces which can be reassembled to yield the second. Obviously, any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.
For every polyhedron P, Dehn defines a value, now known as the Dehn invariant D(P), with the following property:
From this it follows
and in particular
He then shows that every cube
has Dehn invariant zero while every regular tetrahedron
has non-zero Dehn invariant. This settles the matter.
A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. Note that if a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths. Similarly, if a polyhedron is cut along an edge, the corresponding angle is cut into two. However, normally cutting a polyhedron introduces new edges and angles; we need to make sure that the contributions of these cancel out. The two angles introduced will always add up to π
; we therefore define our Dehn invariant so that multiples of angles of π give a net contribution of zero.
All of the above requirements can be met if we define D(P) as an element of the tensor product
of the real number
s R and the quotient space
R/(Qπ) in which all rational multiples of π are zero. For the present purposes, it suffices to consider this as a tensor product of Z-modules (or equivalently of abelian groups). However, the more difficult proof of the converse (see below) makes use of the vector space
structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.
Let ℓ(e) be the length of the edge e and θ(e) be the dihedral angle
between the two faces meeting at e, measured in radian
s. The Dehn invariant is then defined as
where the sum is taken over all edges e of the polyhedron P.
(1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant. Børge Jessen
later extended Sydler's results to four dimensions. In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology
of certain classical group
s.
Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space
can be tiled
periodically is zero.
T1 and T2 with equal base area and equal height (and therefore equal volume), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to T1 and also glued to T2, the resulting polyhedra are scissors-congruent?
Dehn's invariant can be used to yield a negative answer also to this stronger question.
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
of equal 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....
, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, Hilbert conjectured that this is not always possible. This was confirmed within the year by his student 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...
, who proved that the answer in general is "no" by producing a counterexample.
The answer for the analogous question about 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 in 2 dimensions is "yes" and had been known for a long time; this is the Bolyai–Gerwien theorem
Bolyai–Gerwien theorem
In geometry, the Bolyai–Gerwien theorem, named after Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e...
.
History and motivation
The formula for the volume of a pyramidPyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
,
had been known to Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, but all proofs of it involve some form of limiting process
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
or calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, notably the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...
or, in more modern form, Cavalieri's principle
Cavalieri's principle
In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...
. Similar formulas in plane geometry can be proven with more elementary means. Gauss regretted this defect in two of his letters. This was the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods? Because if not, then an elementary proof of Euclid's result is also impossible.
Dehn's answer
Dehn's proof is an instance in which abstract algebraAbstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
is used to prove an impossibility result in geometry
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 ....
. Other examples are doubling the cube
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
and trisecting the angle.
We call two polyhedra scissors-congruent if the first can be cut into finitely many polyhedral pieces which can be reassembled to yield the second. Obviously, any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.
For every polyhedron P, Dehn defines a value, now known as the Dehn invariant D(P), with the following property:
- If P is cut into two polyhedral pieces P1 and P2 with one plane cut, then D(P) = D(P1) + D(P2).
From this it follows
- If P is cut into n polyhedral pieces P1,...,Pn, then D(P) = D(P1) + ... + D(Pn)
and in particular
- If two polyhedra are scissors-congruent, then they have the same Dehn invariant.
He then shows that every cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
has Dehn invariant zero while every regular tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
has non-zero Dehn invariant. This settles the matter.
A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. Note that if a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths. Similarly, if a polyhedron is cut along an edge, the corresponding angle is cut into two. However, normally cutting a polyhedron introduces new edges and angles; we need to make sure that the contributions of these cancel out. The two angles introduced will always add up to π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
; we therefore define our Dehn invariant so that multiples of angles of π give a net contribution of zero.
All of the above requirements can be met if we define D(P) as an element of the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R and the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
R/(Qπ) in which all rational multiples of π are zero. For the present purposes, it suffices to consider this as a tensor product of Z-modules (or equivalently of abelian groups). However, the more difficult proof of the converse (see below) makes use of the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.
Let ℓ(e) be the length of the edge e and θ(e) be the dihedral angle
Dihedral angle
In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
between the two faces meeting at e, measured in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s. The Dehn invariant is then defined as
where the sum is taken over all edges e of the polyhedron P.
Further information
In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? SydlerJean-Pierre Sydler
Jean-Pierre Sydler is a Swiss mathematician and a librarian, well known for his work in geometry, most notably on Hilbert's third problem.-Biography :...
(1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant. Børge Jessen
Børge Jessen
Børge Christian Jessen was a Danish mathematician best known for his work in analysis, specifically on zeta function, and in geometry, specifically on Hilbert's third problem....
later extended Sydler's results to four dimensions. In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
of certain classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s.
Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
can be tiled
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
periodically is zero.
Original question
Hilbert's original question was more complicated: given any two tetrahedraTetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
T1 and T2 with equal base area and equal height (and therefore equal volume), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to T1 and also glued to T2, the resulting polyhedra are scissors-congruent?
Dehn's invariant can be used to yield a negative answer also to this stronger question.