View source for Hilbert's third problem

{{Short description|On dissections between polyhedra}}
[[File:Cube and prism from two bricks.svg|thumb|Two polyhedra of equal volume, cut into two pieces which can be reassembled into either polyhedron]]
The third of [[Hilbert's problems|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 [[polyhedron|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 [[Carl Friedrich Gauss]],<ref>[[Carl Friedrich Gauss]]: ''Werke'', vol. 8, pp. 241 and 244</ref> [[David 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.<ref>{{cite journal |first=Max |last=Dehn |title=Ueber den Rauminhalt |journal=[[Mathematische Annalen]] |volume=55 |year=1901 |issue=3 |pages=465–478 |doi=10.1007/BF01448001 |s2cid=120068465 |url=https://zenodo.org/record/2327856 }}</ref>

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 [[Wallace–Bolyai–Gerwien theorem]].

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]], and was solved by [[Ludwik Birkenmajer|Ludwik Antoni Birkenmajer]] with a different method than Dehn's. Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.<ref name=":0">{{Cite journal|last1=Ciesielska|first1=Danuta|last2=Ciesielski|first2=Krzysztof|date=2018-05-29|title=Equidecomposability of Polyhedra: A Solution of Hilbert's Third Problem in Kraków before ICM 1900|journal=The Mathematical Intelligencer|volume=40|issue=2|pages=55–63|language=en|doi=10.1007/s00283-017-9748-4|issn=0343-6993|doi-access=free}}</ref>

==History and motivation==
The formula for the volume of a [[pyramid (geometry)|pyramid]], one-third of the product of base area and height, had been known to [[Euclid]]. Still, all proofs of it involve some form of [[Limit of a sequence|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 to [[Christian Ludwig Gerling]], who proved that two symmetric tetrahedra are [[equidecomposable]].<ref name=":0" />

Gauss's 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 proof ==
{{main article|Dehn invariant}}
Dehn's proof is an instance in which [[abstract algebra]] is used to prove an impossibility result in [[geometry]]. Other examples are [[doubling the cube]] and [[trisecting the angle]].

Two polyhedra are called {{anchor|1=Scissors congruence|2=Scissors-congruent}}''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 (logic)|converse]].

For every polyhedron <math>P</math>, Dehn defines a value, now known as the [[Dehn invariant]] <math>\operatorname{D}(P)</math>, with the property that,
if <math>P</math> is cut into polyhedral pieces <math>P_1, P_2, \dots P_n</math>, then
<math display=block>\operatorname{D}(P) = \operatorname{D}(P_1)+\operatorname{D}(P_2)+\cdots + \operatorname{D}(P_n).</math>
In particular, if two polyhedra are scissors-congruent, then they have the same Dehn invariant. He then shows that every [[cube]] has Dehn invariant zero while every regular [[tetrahedron]] 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 <math>\pi</math>, and the angles introduced around an edge interior to the polyhedron add to <math>2\pi</math>. Therefore, the Dehn invariant is defined in such a way that integer multiples of angles of <math>\pi</math> give a net contribution of zero.

All of the above requirements can be met by defining <math>\operatorname{D}(P)</math> as an element of the [[tensor product]] of the [[real number]]s <math>\R</math> (representing lengths of edges) and the [[Quotient space (linear algebra)|quotient space]] <math>\R/(\Q\pi)</math> (representing angles, with all rational multiples of <math>\pi</math> replaced by zero).<ref name=eom>{{SpringerEOM|first=M.|last=Hazewinkel|authorlink= Michiel Hazewinkel |title=Dehn invariant|id=Dehn_invariant&oldid=35803}}</ref> For some purposes, this definition can be made using the [[tensor product of modules]] over <math>\Z</math> (or equivalently of [[abelian group]]s), while other aspects of this topic make use of a [[vector space]] structure on the invariants, obtained by considering the two factors <math>\R</math> and <math>\R/(\Q\pi)</math> to be vector spaces over <math>\Q</math> and taking the [[Tensor product|tensor product of vector spaces]] over <math>\Q</math>. 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 <math>e</math> of a polyhedron <math>P</math>, let <math>\ell(e)</math> be its length and let <math>\theta(e)</math> denote the [[dihedral angle]] of the two faces of <math>P</math> that meet at <math>e</math>, measured in [[radian]]s and considered modulo rational multiples of <math>\pi</math>. The Dehn invariant is then defined as
<math display=block>\operatorname{D}(P) = \sum_{e} \ell(e)\otimes \theta(e)</math>
where the sum is taken over all edges <math>e</math> of the polyhedron <math>P</math>.<ref name=eom /> It is a [[Valuation (geometry)|valuation]].

==Further information==
In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? [[Jean-Pierre Sydler|Sydler]] (1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant.<ref>{{cite journal |last=Sydler |first=J.-P. |title=Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions |journal=[[Commentarii Mathematici Helvetici|Comment. Math. Helv.]] |volume=40 |year=1965 |pages=43–80 |doi= 10.1007/bf02564364|s2cid=123317371 }}</ref>  [[Børge Jessen]] later extended Sydler's results to four dimensions.<ref>{{cite journal
 | last = Jessen | first = Børge
 | journal = Nachrichten der Akademie der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Fachgruppe II: Nachrichten aus der Physik, Astronomie, Geophysik, Technik
 | mr = 353150
 | pages = 47–53
 | title = Zur Algebra der Polytope
 | year = 1972
 | zbl = 0262.52004}}</ref>  In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the [[homology (mathematics)|homology]] of certain [[classical group]]s.<ref>{{cite journal |first1=Johan |last1=Dupont |first2=Chih-Han |last2=Sah |title=Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences |journal=[[Acta Mathematica|Acta Math.]] |volume=164 |year=1990 |issue=1–2 |pages=1–27 |doi=10.1007/BF02392750 |doi-access=free }}</ref>

Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of [[three-dimensional space]] can be [[honeycomb (geometry)|tiled]] periodically is zero.<ref>{{cite journal |first=Hans E. |last=Debrunner |title=Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln |journal=[[Archiv der Mathematik|Arch. Math.]] |volume=35 |year=1980 |issue=6 |pages=583–587 |doi=10.1007/BF01235384 |s2cid=121301319 }}</ref>

{{unsolved|mathematics|In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?}}
Jessen also posed the question of whether the analogue of Jessen's results remained true for [[spherical geometry]] and [[hyperbolic geometry]]. 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.<ref>{{citation
 |last        = Dupont
 |first       = Johan L.
 |doi         = 10.1142/9789812810335
 |isbn        = 978-981-02-4507-8
 |mr          = 1832859
 |page        = 6
 |publisher   = World Scientific Publishing Co., Inc., River Edge, NJ
 |series      = Nankai Tracts in Mathematics
 |title       = Scissors congruences, group homology and characteristic classes
 |url         = http://home.math.au.dk/dupont/scissors.ps
 |volume      = 1
 |year        = 2001
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20160429152252/http://home.math.au.dk/dupont/scissors.ps
 |archive-date = 2016-04-29
}}.</ref>

==Original question==
Hilbert's original question was more complicated: given any two [[tetrahedron|tetrahedra]] ''T''<sub>1</sub> and ''T''<sub>2</sub> 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''<sub>1</sub> and also glued to ''T''<sub>2</sub>, 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]]
* [[Onorato Nicoletti]]

==References==
{{Reflist}}

==Further reading==
*{{cite journal |first=D. |last=Benko |title=A New Approach to Hilbert's Third Problem | journal=[[The American Mathematical Monthly]] |volume=114 |issue=8 |year=2007 |pages=665–676 |doi=10.1080/00029890.2007.11920458|s2cid=7213930 }}
*{{cite web|first=Rich |last=Schwartz |url=http://www.math.brown.edu/~res/Papers/dehn_sydler.pdf |title=The Dehn–Sydler Theorem Explained |year=2010 }}
*{{cite book|last1=Koji|first1=Shiga|last2=Toshikazu Sunada|author2-link=Toshikazu Sunada|title=A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra|publisher=American Mathematical Society|year=2005}}

==External links==
*[http://everything2.com/e2node/Proof%2520for%2520Hilbert%2527s%2520third%2520problem Proof of Dehn's Theorem at Everything2]
*{{MathWorld |id=DehnInvariant |title=Dehn Invariant}}
*[http://everything2.com/e2node/Dehn%2520invariant Dehn Invariant at Everything2]
*{{SpringerEOM| title=Dehn invariant | id=Dehn_invariant | oldid=13481 | first=M. | last=Hazewinkel }}

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[[Category:Hilbert's problems|#03]]
[[Category:Euclidean solid geometry]]
[[Category:Geometric dissection]]
[[Category:Geometry problems]]