  A Kokotsakis polyhedron with quadrangular base is a neighborhood of a
quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid.
Up to now, several flexible classes were known, but a complete classification
was missing. In the present paper, we provide such a classification. The
analysis is based on the fact that the dihedral angles of a Kokotsakis
polyhedron are related by an Euler-Chasles correspondence. It results in a
diagram of elliptic curves covering complex projective planes. A polyhedron is
flexible if and only if this diagram commutes.
