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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 (this is also true in higher dimensions).
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 .
== 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 .
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