Scissors congruence

The third on 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 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 Gauss, 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.

The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Bolyai–Gerwien theorem.

The formula for the volume of a pyramid,

had been known to Euclid, but all proofs of it involve some form of 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. This was 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 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.

We call two polyhedra scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Obviously, any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.

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