Abstract: Hilbert's third problem, originally duo to Gauss, asks: "Does there exist two tetrahedra with equal (namely equiarea) bases and equal heights which can in no way be split up into finitely many congruent tetrahedra, and which can not be combined with finitely many congruent tetrahedra to form two polyhedra which themselves could be split up into finitely many congruent tetrahedra?" Gauss was interested in this problem because if the answer were NO it would give an elementary proof (avoiding continuity arguments like exhaustion principle) for the antiquity proposition that two tetrahedra with equal bases and heights have the same volume. Unfortunately, the answer is YES, as first shown by Max Dehn in 1902. His arguments has simplified a great deal through the years, and we present a simple elementary proof duo to David Benko discovered in 2007.
Host: Xiang Tang