Editing Polyhedron (section)

===Dehn invariant===
{{main|Dehn invariant}}
In two dimensions, the [[Bolyai–Gerwien theorem]] asserts that any polygon may be transformed into any other polygon of the same area by [[Dissection problem|cutting it up into finitely many polygonal pieces and rearranging them]].  The analogous question for polyhedra was the subject of [[Hilbert's third problem]]. [[Max Dehn]] solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the [[Dehn invariant]], such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other.<ref>{{citation |last=Sydler |first=J.-P. | author-link = Jean-Pierre Sydler |title=Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions |journal=[[Commentarii Mathematici Helvetici|Comment. Math. Helv.]]|language=fr |volume=40 |year=1965 |pages=43–80 |doi= 10.1007/bf02564364| mr = 0192407|s2cid=123317371 | url = https://eudml.org/doc/139296 }}</ref> The Dehn invariant is not a number, but a [[Vector (mathematics)|vector]] in an infinite-dimensional vector space, determined from the lengths and [[dihedral angle]]s of a polyhedron's edges.<ref>{{SpringerEOM|first=M.|last=Hazewinkel|title=Dehn invariant|id=Dehn_invariant&oldid=35803}}</ref>

Another of Hilbert's problems, [[Hilbert's 18th problem|Hilbert's eighteenth problem]], concerns (among other things) polyhedra that [[Honeycomb (geometry)|tile space]]. Every such polyhedron must have Dehn invariant zero.<ref>{{citation
 | last = Debrunner | first = Hans E.
 | doi = 10.1007/BF01235384
 | issue = 6
 | journal = [[Archiv der Mathematik]]
 | language = de
 | mr = 604258
 | pages = 583–587
 | title = Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln
 | volume = 35
 | year = 1980| s2cid = 121301319
 }}.</ref> The Dehn invariant has also been connected to [[flexible polyhedron|flexible polyhedra]] by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.<ref>{{citation
 | last = Alexandrov | first = Victor
 | arxiv = 0901.2989
 | doi = 10.1007/s00022-011-0061-7
 | issue = 1–2
 | journal = Journal of Geometry
 | mr = 2823098
 | pages = 1–13
 | title = The Dehn invariants of the Bricard octahedra
 | volume = 99
 | year = 2010| citeseerx = 10.1.1.243.7674
 | s2cid = 17515249
 }}.</ref>