Flexible polyhedral surfaces with two flat poses (Q2406215)

Summary: 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.