David Gale wrote: << Dehn only proved the necessity. The sufficiency was proved (?) seventy years later by a Danish mathematician named Sydler and a clear proof was given by Borge Jessen shortly afterwards. The proof is difficult and uses homological algebra (!). The book "Hilbert's Third Problem" by Vladimir Boltyanskii 1978 is the definitive reference. ...

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I have that book and it's very nice, written at a level most upper division math undergrads could understand, though I don't think he tries to prove Sydler's sufficiency proof. QUESTION: I've never heard of any generalization of Dehn-Sydler to higher dimensions than 3. Does anyone out there know of progress in higher dimensions? --Dan Asimov