Hilbert's third problem
A regular tetrahedron and a cube have the same volume. Can the tetrahedron be cut into finitely many polyhedral pieces that can be reassembled to form the cube? Prove your answer.
History and references
Solved in 1900 by Max Dehn.
In the plane, any two polygons of equal area can be cut into each other, by the Wallace–Bolyai–Gerwien theorem. Gauss had regretted that the volume of a pyramid seemed to need limiting arguments, and in his 1900 list Hilbert asked whether cutting and regluing suffices in three dimensions. His student Max Dehn answered no in the same year, making it the first of Hilbert's problems to be solved. Dehn attached to each polyhedron an invariant, built from its edge lengths and dihedral angles, that does not change under cutting and regluing. A cube has invariant zero and a regular tetrahedron does not, so no such dissection exists. In 1965 Jean-Pierre Sydler showed that equal volume together with equal Dehn invariant is also sufficient.
References
Solutions 0
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