The first examples of flexible polyhedra, now called Bricard's octahedra, were discovered by . They are self-intersecting surfaces isometric to an octahedron. The first example of a non-self-intersecting surface in R3, the Connelly sphere, was discovered by .
In the late 1970's Connelly and D. Sullivan formulated the Bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by .
Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. If this is true then it would lead to a scissor congruence under flexing for the volume together with the Dehn invariant. The special case of mean curvature has been proved by Ralph Alexander.