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.
Polyhedra can bend but not breathe. (new mathematical proof shows that polyhedra must keep their volume constant while moving)
Mar 13, 1998; Anyone who has made an origami crane knows the delight and wonderment of conjuring a moving creature from the static geometry of...