Hilbert's Third Problem
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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 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 t ...
of equal
volume Volume is a measure of occupied 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). The de ...
, 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
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
conjectured that this is not always possible. This was confirmed within the year by his student
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. ...
, who proved that the answer in general is "no" by producing a counterexample. The answer for the analogous question about
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 toge ...
s in 2 dimensions is "yes" and had been known for a long time; this is the
Wallace–Bolyai–Gerwien theorem In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and Paul Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it ...
. Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of
Kraków Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...
, and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn. Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.


History and motivation

The formula for the volume of a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
, :\frac, had been known to
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, but all proofs of it involve some form of limiting process or
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, notably the
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 bet ...
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 to
Christian Ludwig Gerling Christian Ludwig Gerling (10 July 1788 – 15 January 1864) studied under Carl Friedrich Gauss, obtaining his doctorate in 1812 for a thesis entitled: ''Methodi proiectionis orthographicae usum ad calculos parallacticos facilitandos explicavit ...
, who proved that two symmetric tetrahedra are equidecomposable. Gauss' letters were 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 algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
is used to prove an impossibility result 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 c ...
. Other examples are
doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
and
trisecting the angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...
. Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Any two scissors-congruent polyhedra have the same volume. Hilbert asks about the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
. For every polyhedron P, Dehn defines a value, now known as the
Dehn invariant In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled (" dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who us ...
\operatorname(P), with the property that, if P is cut into polyhedral pieces P_1, P_2, \dots P_n, then \operatorname(P) = \operatorname(P_1)+\operatorname(P_2)+\cdots + \operatorname(P_n). In particular, if two polyhedra are scissors-congruent, then they have the same Dehn invariant. He then shows that every
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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
has Dehn invariant zero while every regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
has non-zero Dehn invariant. Therefore, these two shapes cannot be scissors-congruent. A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. 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. Cutting a polyhedron typically also introduces new edges and angles; their contributions must cancel out. The angles introduced when a cut passes through a face add to \pi, and the angles introduced around an edge interior to the polyhedron add to 2\pi. Therefore, the Dehn invariant is defined in such a way that integer multiples of angles of \pi give a net contribution of zero. All of the above requirements can be met by defining \operatorname(P) as an element of the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R (representing lengths of edges) and the quotient space \R/(\Q\pi) (representing angles, with all rational multiples of \pi replaced by zero). For some purposes, this definition can be made using the
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produc ...
over \Z (or equivalently of
abelian groups In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
), while other aspects of this topic make use of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
structure on the invariants, obtained by considering the two factors \R and \R/(\Q\pi) to be vector spaces over \Q and taking the
tensor product of vector spaces In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
over \Q. This choice of structure in the definition does not make a difference in whether two Dehn invariants, defined in either way, are equal or unequal. For any edge e of a polyhedron P, let \ell(e) be its length and let \theta(e) denote the
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
of the two faces of P that meet at e, measured in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s and considered modulo rational multiples of \pi. The Dehn invariant is then defined as \operatorname(P) = \sum_ \ell(e)\otimes \theta(e) where the sum is taken over all edges e of the polyhedron P. It is a valuation.


Further information

In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? Sydler (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 Christian Jessen (19 June 1907 – 20 March 1993) was a Denmark, Danish mathematician best known for his work in Mathematical analysis, analysis, specifically on the Riemann zeta function, and in geometry, specifically on Hilbert's third pr ...
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 may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
of certain
classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or s ...
s. Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
can be tiled periodically is zero. Jessen also posed the question of whether the analogue of Jessen's results remained true for spherical geometry and
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
. In these geometries, Dehn's method continues to work, and shows that when two polyhedra are scissors-congruent, their Dehn invariants are equal. However, it remains an open problem whether pairs of polyhedra with the same volume and the same Dehn invariant, in these geometries, are always scissors-congruent..


Original question

Hilbert's original question was more complicated: given any two
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
''T''1 and ''T''2 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 ''T''1 and also glued to ''T''2, the resulting polyhedra are scissors-congruent? Dehn's invariant can be used to yield a negative answer also to this stronger question.


See also

*
Hill tetrahedron In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M. J. M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube. Const ...
*
Onorato Nicoletti Onorato Nicoletti (21 June 1872, Rieti – 31 December 1929, Pisa) was an Italian mathematician. Biography Nicoletti received his ''laurea'' in 1894 from the Scuola Normale di Pisa. In 1898, he became a professor of infinitesimal calculus at th ...


References


Further reading

* * *


External links


Proof of Dehn's Theorem at Everything2
*
Dehn Invariant at Everything2
* {{Authority control #03 Euclidean solid geometry Geometric dissection