Rewriting the rules of kirigami

All origami artists should have in their repertoire certain basic folds—the petal, the crimp, the rabbit ear, to name just a few. Skillfully deployed, those basic folds can serve as building blocks for intricate, original creations. Now Randall Kamien, Shu Yang, and coworkers at the University of Pennsylvania have used concepts from condensed-matter theory to identify fundamental elements in a related art form: kirigami, the art of cutting, folding, and pasting. ¹

The Penn group specifically considered kirigami on a honeycomb lattice—a choice inspired by the prospect of fashioning three-dimensional nanostructures from self-assembled DNA networks and from graphene and graphene-like materials. The researchers impose constraints designed to mimic the bond networks in those materials: Cuts that remove part of a lattice are allowed, for instance, but only if subsequent folding and bond reformation around the excised area can restore the lattice’s connectivity without altering the bond lengths.

Panel a in the figure shows one of the group’s prototypical kirigami templates. The hole at the center can be closed only with a specific set of folds—along the dotted lines—that cause parts of the surface to pop out of the plane, as shown in panel b. So-called mountain folds (M) mark the perimeters of the raised plateaus; valley folds (V) mark the perimeter of the basin.

Kamien and his colleagues find that cut-and-fold sequences can be characterized by the topological defects they leave behind. Any sequence giving rise to a 3D feature will necessarily create disclinations, sites at which the lattice symmetry is disrupted. In the figure, for instance, the mountain and valley folds converge on pairs of cells having five (yellow) and seven (green) neighbors, instead of the usual six. Known by condensed-matter theorists as Stone–Wales defects, those pairs and their associated folds are encoded by V-shaped cuts—dubbed five–seven cuts and highlighted in blue in panel a—that each bisect adjacent edges on the lattice.

While experimenting with paper templates, Xingting Gong, an undergraduate student in Kamien’s lab, discovered a second kind of kirigami cut that creates a pair of vertices having two and four neighbors instead of three. That so-called two–four cut, also V-shaped, runs along adjacent edges instead of bisecting them. The resulting plateau is taller and more sharply angled than that of the five–seven cut.

Kamien’s postdoc Toen Castle deduced that the two–four and five–seven cuts are the fundamental building blocks from which all allowable kirigami structures derive. In essence, that’s because folds can preserve lattice distances only if the angle of the associated cut is an integer multiple of an angle of rotational symmetry. The two–four and five–seven cuts are, respectively, the minimum-angle cuts satisfying that criterion for the honeycomb’s three- and sixfold rotational symmetries. By cleverly mixing and matching those cuts, one can design kirigami templates that fold into boxes, staircases, and other shapes.

In real systems, if the energy gained by repairing a lattice’s bonds exceeds the energy costs of folding it, a kirigami template can be self-assembling. Recent simulations suggest that kirigami might be useful for fashioning self-assembling graphene containers for hydrogen storage or for manipulating mechanical and electronic properties in graphene-like materials. ²

Kamien thinks, however, that from a technical standpoint kirigami might be more easily applied to systems having larger length scales, such as DNA assemblies. For instance, DNA origami can be used to shape genetic material into tiles that then self-assemble into 2D periodic lattices. ³ Kirigami could provide a means to coax those lattices into drug delivery capsules, nanomachine parts, and other 3D structures.

Because unwanted material is snipped away at the outset, kirigami in principle permits the fabrication of more complicated structures—from fewer, simpler folds—than can be made with origami alone. The Penn group is now working to devise algorithms to exploit that feature. Says Kamien, “How do you make something that pops up into, say, a pyramid, using as little material as possible? Now that we have a basic framework, we can begin tackling those kinds of questions.”