Flexible polyhedron

Example

A flexible polyhedron is a closed polyhedral surface in $\R^3$ that admits a continuous non-rigid motion in which every face stays congruent to itself. Its edges act as hinges, and the shape changes while each face moves rigidly. In the language of rigidity of frameworks, its framework of vertices and edges, with the faces triangulated, is flexible.

Bricard found flexible octahedra in 1897. Their surfaces pass through themselves, so they are not boundaries of solids. Connelly constructed in 1977 the first flexible polyhedron without self-intersections. A simpler example is Steffen's polyhedron, with $9$ vertices, $21$ edges and $14$ triangular faces. It satisfies Euler's formula $9 - 21 + 14 = 2$.

These examples are not convex, as they must be. Cauchy's rigidity theorem shows that no convex polyhedron is flexible.