A Classification of Flexible Kokotsakis Polyhedra with Reducible Quadrilaterals

TL;DR

This paper classifies flexible Kokotsakis polyhedra with quadrilaterals that have reducible polynomial relations, expanding understanding of conditions under which these structures can flex.

Contribution

It provides a complete classification of flexible Kokotsakis polyhedra with reducible quadrilaterals, a problem previously largely unsolved.

Findings

Identifies algebraic conditions for reducibility of dihedral angle relations.

Characterizes shape restrictions leading to flexibility.

Completes classification of a special class of flexible polyhedra.

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.