Flexible polyhedron


In geometry, a flexible polyhedron is a polyhedral surface that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex.
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 flexible non-self-intersecting surface in R3, the Connelly sphere, was discovered by. Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra.

Bellows conjecture

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
using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. 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. Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously.

Scissor congruence

Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This was known as the strong bellows conjecture or the strong bellows theorem.
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.

Generalizations

Flexible 4-polytopes in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by. In dimensions, flexible polytopes were constructed by.

Primary sources