Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type

Abstract

A. 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 (3 × 3 ) building block of a so-called V-hedra composed of skew quads. The latter also gives a positive answer to a question posed by R. Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist.