A kinematic approach to Kokotsakis meshes

A Kokotsakis mesh is a polyhedral structure consisting of an n-sided central polygon P-0 surrounded by a belt of quadrangles or triangles in the following way: Each side a(i) of P-0 is shared by an adjacent polygon P-i, and the relative motion between cyclically consecutive neighbor polygons is a spherical coupler motion. Hence, each vertex of P-0 is the meeting point of four faces. In the case n = 3 the mesh is part of an octahedron. These structures with rigid faces and variable dihedral angles were first studied in the thirties of the last century. However, in the last years there was a renaissance: The question under which conditions such meshes are infinitesimally or continuously flexible gained high actuality in discrete differential geometry. The goal of this paper is to revisit the well-known continuously flexible examples (Bricard, Graf, Sauer, Kokotsakis) from the kinematic point of view and to extend their list by a new family. (C) 2010 Elsevier B.V. All rights reserved.