What is... a scissors congruence

Two polyhedra in Euclidean 3-space are called scissors congruent (s.c.) if they can be subdivided into the same finite number of smaller polyhedra such that each piece in the first polyhedron is congruent to one in the second. If two polyhedra are s.c., then they clearly have the same volume; and for the analogous notion of s.c. in the plane, it was probably known already by the Greeks that two polygons are s.c. if and only if they have the same area. However, based on some remarks in a letter by C. F. Gauss (1844), D. Hilbert included on his famous list of mathematical problems (1900) the question of finding two polyhedra of the same volume that could be proven not to be s.c. This is Hilbert’s 3rd problem, and it was solved by M. Dehn (1901), who found a necessary condition for two polyhedra to be s.c. that he showed was not satisfied for the cube and the regular tetrahedron of the same volume. Finally, J. P. Sydler (1965) showed that equal volume together with Dehn’s condition are also sufficient for s.c. of two polyhedra in Euclidean 3-space. The notion of s.c. of polytopes makes sense in all dimensions, as well as in spherical or even hyperbolic geometry. As a model for hyperbolic n-spaceH n we use the upper half space inR, consisting of points whose last coordinate is positive, and in this model hyperbolic lines, planes, etc., are Euclidean half circles (or lines), half spheres (or half planes), etc., perpendicular to the boundary. Now a polytope in Euclidean (R), spherical (S), or hyperbolic n-space (H ) is a compact body that can be decomposed into finitely many simplices.