A Classification of Flexible Kokotsakis Polyhedra with Reducible Quadrilaterals

Abstract: 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. In contrast to existing work, the quadrilateral faces do not necessarily have to be planar. In general, such a mesh is rigid. 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, this article deals with the case when these polynomials are reducible. We explore the conditions for reducibility to characterize all possible shape restrictions that lead to flexible polyhedra.