Are the Universal Molecules Foldable?

Lang's Universal Molecule algorithm produces crease patterns for which a folded uniaxial base realization exists. A question remains, however, about whether they are indeed reachable by a folding process. We restrict the discussion to panel-and-hinge style folding processes which rigidly maintain the faces rather than allowing bending or stretching of the paper. This question is important, for instance, for the development of deployable structures, such as a foldable satellite solar array, since such structures are often best modeled as a panel-and-hinge framework. The figure below illustrates such a folding process. In this figure, a simple crease pattern is rigidly folded to a uniaxial base realization.

A rigid folding process.

It turns out that many of the universal molecules produced by Lang's algorithm are not reachable by rigid folding processes. The figure below is an example of a crease pattern for which there exists a realization as a uniaxial base, but no folding process exists to reach the base from the flat state if the origami is treated as a panel and hinge structure.

Furthermore, the flat state itself is completely rigid, meaning that no non-trivial realization of the crease pattern can be reached by rigid folding.

A completely rigid universal molecule.