Dehn’s Dissection Theorem

such that the smaller polyhedra have pairwise disjoint interiors. Note that there is not an additional assumption, say, that the smaller polyhedra meet face to face. Two polyhedra P and Q are scissors congruent if there are dissections P = P1 ∪ ...∪Pn and Q = Q1 ∪ ...∪Qn such that each Pk is isometric to Qk. Sometimes, one requires that all the isometries are orientation-preserving, but in fact and two shapes that are scissors congruent via general isometries are also scissors congruent via orientation preserving isometries. (This little fact isn’t something that is important for our purposes.) One could make the same definition for polygons, and any two polygons of the same area are scissors congruent to each other. (J. Dupont says that W. Wallace first formally proved this in 1807.) Hilbert asked if every two polyhedra of the same volume are scissors congruent to each other. In 1901, Max Dehn proved the now-famous result that the cube and the regular tetrahedron (of the same volume) are not scissors congruent. We’re going to give a proof of this result.