The third of 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 Carl Friedrich Gauss,[1] 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.[2]
The answer for the analogous question about polygons 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 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.[3]
- ^ Carl Friedrich Gauss: Werke, vol. 8, pp. 241 and 244
- ^ Dehn, Max (1901). "Ueber den Rauminhalt". Mathematische Annalen. 55 (3): 465–478. doi:10.1007/BF01448001. S2CID 120068465.
- ^ Ciesielska, Danuta; Ciesielski, Krzysztof (2018-05-29). "Equidecomposability of Polyhedra: A Solution of Hilbert's Third Problem in Kraków before ICM 1900". The Mathematical Intelligencer. 40 (2): 55–63. doi:10.1007/s00283-017-9748-4. ISSN 0343-6993.