Generalizing continuous flexible Kokotsakis belts of the isogonal type

TL;DR

This paper extends the study of flexible Kokotsakis belts by allowing non-planar faces and focusing on isogonal conditions, demonstrating the existence of continuous flexible skew quad surfaces.

Contribution

It generalizes classical Kokotsakis belts to include skew faces under isogonal constraints, addressing a question about flexible skew quad surfaces.

Findings

Established the existence of continuous flexible skew quad surfaces.

Extended the theory of Kokotsakis belts to non-planar faces.

Provided a positive answer to Sauer's 1970 question.

Abstract

Kokotsakis studied the following problem in 1932: Given is a rigid closed polygonal line (planar or non-planar), which is surrounded by a polyhedral strip, where at each polygon vertex three faces meet. Determine the geometries of these closed strips with a continuous mobility. On the one side, we generalize this problem by allowing the faces, which are adjacent to polygon line-segments, to be skew; i.e to be non-planar. But on the other side, we restrict to the case where the four angles associated with each polygon vertex fulfill the so-called isogonality condition that both pairs of opposite angles are equal or supplementary. In more detail, we study the case where the polygonal line is a skew quad, as this corresponds to a (3x3) building block of a so-called V-hedra composed of skew quads. The latter also gives a positive answer to a question posed by Robert Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist.