Steffen's polyhedron is able to flex without self-intersection and without distorting any of its faces. It has 14 triangular faces, 21 edges, and 9 vertices. Euler had conjectured no polyhedron could flex. Cauchy proved that convex polyhedra cannot flex.
The bellows conjecture (proven in 1997) asserts that a flexible polyhedron has constant volume as it is flexed.