Editing Hilbert's third problem (section)

==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>