Flexible polyhedron

The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by Raoul Bricard (1897). They are self-intersecting surfaces isometric to an octahedron. The first example of a flexible non-self-intersecting surface in ${\displaystyle {\mathbb{R}}^3}$, the Connelly sphere, was discovered by Robert Connelly (1977). 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 I. Kh. Sabitov (1995) using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by Robert Connelly, I. Sabitov, and Anke Walz (1997). 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.