Kokotsakis Flexible Polyhedra: Generalization of the Orthodiagonal Involutive Type

In this paper, we introduce and study a remarkable class of mechanisms formed by a $3\times 3$ arrangement of rigid and skew quadrilateral faces with revolute joints at the common edges. These Kokotsakis-type mechanisms with a quadrangular base and non-planar faces are a generalization of Izmestiev's orthodiagonal involutive type of Kokotsakis polyhedra formed by planar quadrilateral faces. The importance of this class is undisputed as it represents the first known flexible discrete surface -- T-nets -- which has been constructed by Graf and Sauer. Our algebraic approach yields a complete characterization of all complexes of the orthodiagonal involutive type. It is shown that one has 8 degrees of freedom to construct such mechanisms. This is illustrated by several examples including cases which could not be realized using planar faces. We demonstrate the practical realization of the proposed mechanisms by building a physical prototype using stainless steel. This is illustrated in our video https://youtu.be/NU7P15v2LaQ which can be viewed on YouTube. In contrast to plastic prototype fabrication, we avoid large tolerances and inherent flexibility.