Flexible polyhedron

In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. 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 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.^[1]