Flexible polyhedron

A flexible polyhedron is a polyhedron that, if physically constructed with rigid plates for faces and hinges as edges, can deform in shape. A polyhedron that cannot deform at all is known as rigid, and a polyhedron that can deform but only infinitesimally is known as shaky. All convex polyhedra are rigid, as proven by Cauchy. The bellows conjecture, proven in 1997, states that the volume of a flexible polyhedron is constant as it is deformed; however, counterexamples to the bellows conjecture have been found in non-Euclidean space.

Bricard found a family of flexible octahedra in 1897 known as Bricard octahedra, but they are self-intersecting. In 1978, Connelly found the first physically realizable flexible polyhedron, with 18 triangular faces. Klaus Steffen found a 14-face flexible polyhedron made of only triangles, which has been proven minimal for a polyhedron with only triangular faces.

Most known flexible polyhedra have a single degree of freedom. In 2016, Lijingjiao et al. found a polyhedron with two degrees of freedom.