Flexible Polyhedral Surfaces with Two Flat Poses

We present three types of polyhedral surfaces, which are continuously flexible and have not only an initial pose, where all faces are coplanar, but pass during their self-motion through another pose with coplanar faces (flat pose). These surfaces are examples of so-called rigid origami, since we only admit exact flexions, i.e., each face remains rigid during the motion; only the dihedral angles vary. We analyze the geometry behind Miura-ori and address Kokotsakis' example of a flexible tessellation with the particular case of a cyclic quadrangle. Finally, we recall Bricard's octahedra of Type 3 and their relation to strophoids.