Journal for Geometry and Graphics 26 (2022), No. 2, 237--251
Copyright Heldermann Verlag 2022
On Continuous Flexible Kokotsakis Belts of the Isogonal Type and V-Hedra with Skew Faces
Georg Nawratil
Vienna University of Technology, Vienna, Austria
nawratil@geometrie.tuwien.ac.at
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-hedron composed of skew quads. The latter also gives a partial answer to a question posed by R. Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist.
Keywords: Kokotsakis belt, continuous flexibility, skew quad surface.
MSC: 51N55; 51M04, 51N15
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