Flexible Kokotsakis Meshes with Skew Faces: Generalization of the Orthodiagonal Involutive Type

TL;DR

This paper generalizes a class of flexible quadrilateral mesh mechanisms by removing planarity constraints, providing a complete algebraic characterization, and demonstrating practical physical prototypes with stainless steel.

Contribution

It introduces a broader class of flexible Kokotsakis meshes with skew faces, extending prior planar face models and offering a complete algebraic classification.

Findings

Maximum of 8 degrees of freedom in construction.

Examples include non-planar face mechanisms.

Successful physical prototype realization.

Abstract

In this paper, we introduce and study a remarkable class of mechanisms formed by a $3 \times 3$ arrangement of rigid quadrilateral faces with revolute joints at the common edges. In contrast to the well-studied Kokotsakis meshes with a quadrangular base, we do not assume the planarity of the quadrilateral faces. Our mechanisms are a generalization of Izmestiev's orthodiagonal involutive type of Kokotsakis meshes formed by planar quadrilateral faces. The importance of this Izmestiev 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 flexible $3 \times 3$ quad meshes of the orthodiagonal involutive type up to some degenerated cases. It is shown that one has a maximum of 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. In contrast to plastic prototype fabrication, we avoid large tolerances and inherent flexibility.