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Journal for Geometry and Graphics 16 (2012), No. 2, 153--170
Copyright Heldermann Verlag 2012

A Flexible Planar Tessellation with a Flexion Tiling a Cylinder of Revolution

Hellmuth Stachel
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10/104, 1040 Vienna, Austria

Due to A. Kokotsakis a quad mesh consisting of congruent convex quadrangles of a planar tessellation is flexible. This means, when the flat quadrangles are seen as rigid bodies and only the dihedral angles along internal edges can vary, the mesh admits incongruent realizations in 3-space, so-called flexions. It has recently be proved by the author that at each nontrivial flexion all vertices lie on a cylinder of revolution. In the generic case the complete tessellation is an example of a flexible periodic framework with the property that the symmetry group of each flexion remains isomorphic to that of the initial flat pose.
The goal of this paper is to give a necessary and sufficient condition for a convex quadrangle and for the dihedral angles such that the corresponding flexion forms a tiling on a cylinder, i.e., after bending around the cylinder two boundaries of a finite mesh fit precisely together - apart from a shift. When in such a closing pose the boundaries are glued together along their overlap then the mesh is infinitesimally rigid.

Keywords: Quad mesh, flexible polyhedra, periodic framework, Kokotsakis mesh, cylinder tiling, rigid origami.

MSC: 51M20; 52C25, 53A17, 52B70

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