  Across various scientific and engineering domains, a growing interest in
flexible and deployable structures is becoming evident. These structures
facilitate seamless transitions between distinct states of shape and find broad
applicability ranging from robotics and solar cells to meta-materials and
architecture. In this contribution, we study a class of mechanisms known as
Kokotsakis polyhedra with a quadrangular base. These are $3\times3$
quadrilateral meshes whose faces are rigid bodies and joined by hinges at the
common edges. Compared to prior work, the quadrilateral faces do not have to be
planar. In general, such meshes are not flexible, and the problem of finding
and classifying the flexible ones is old, but until now largely unsolved. It
appears that the tangent values of the dihedral angles between different faces
are algebraically related through polynomials. Specifically, by fixing one
angle as a parameter, the others can be parameterized algebraically and hence
belong to an extended rational function field of the parameter. We use this
approach to characterize shape restrictions resulting in flexible polyhedra.
