Flexible polyhedron

[1] In the late 1970s 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 I. Kh.

The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges.

[3] Because all configurations of a flexible polyhedron have both the same volume and the same Dehn invariant, they are scissors congruent to each other, meaning that for any two of these configurations it is possible to dissect one of them into polyhedral pieces that can be reassembled to form the other.

The total mean curvature of a flexible polyhedron, defined as the sum of the products of edge lengths with exterior dihedral angles, is a function of the Dehn invariant that is also known to stay constant while a polyhedron flexes.