David Hilbert's Problem #3

Hilbert’s Third Problem

Equidecomposability: Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? Equidecomposability

Any polygon can be cut into a finite number of polygonal pieces and reassembled into the shape of any other polygon with the same area. Hilbert’s third problem — the first to be resolved — is whether the same holds for three-dimensional polyhedra. Hilbert’s student Max Dehn answered the question in the negative, showing that a cube cannot be cut into a finite number of polyhedral pieces and reassembled into a tetrahedron of the same volume.

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